Number System is the quietest big scorer in the Punjab Police SI paper: 78 questions since 2016, and not one of them needs a method you could not learn in an evening. Divisibility rules, HCF and LCM, remainders and a handful of fraction facts — that is the whole chapter. The Commission's favourite move is to dress a one-line rule in a story: bells tolling, tiles on a floor, sweets in boxes, students sharing pencils. Learn to spot which of the four tools a story is asking for and the question answers itself. You will learn (1) the divisibility rules worth memorising, (2) HCF and LCM and the one sentence that tells you which to use, (3) how remainders behave, and (4) the fraction and decimal facts the SI repeats.
Divisibility rules
Ten rules. The SI uses them in 27 questions.
You will never be asked to actually divide. You will be shown four numbers and asked which one a given number divides into. With the rules below that is a ten-second question; without them it is a minute of long division you do not have.
| Divisible by | Rule | Quick example |
|---|---|---|
| 2 | Last digit is even | 188822 ✓ |
| 3 | Digit sum divisible by 3 | 81a673: 8+1+a+6+7+3 |
| 4 | Last two digits form a multiple of 4 | 567846 ✗ (46) |
| 5 | Ends in 0 or 5 | 47325 ✓ |
| 6 | Divisible by 2 and 3 | 45236856 ✓ |
| 8 | Last three digits form a multiple of 8 | 8127 ✗ |
| 9 | Digit sum divisible by 9 | 9+9+9+9+0 = 36 ✓ |
| 10 | Ends in 0 | 99990 ✓ |
| 11 | (sum of odd places) − (sum of even places) is 0 or a multiple of 11 | 836: (8+6) − 3 = 11 ✓ |
| 12 | Divisible by 3 and 4 | 45236856 ✓ |
Number the digits from the right: 1st, 2nd, 3rd... Add the odd-place digits, add the even-place digits, subtract. If the answer is 0 or ±11, ±22, ±33..., the number is divisible by 11. For 293Z: odd places are Z and 9 (Z+9), even places are 3 and 2 (5). Z + 9 − 5 = Z + 4 must be 0 or 11, so Z = 7.
- 4 and 8 look at the end, 3 and 9 look at the sum. Mixing those two families is the commonest mistake in the chapter.
- For a composite divisor use coprime factors. 12 = 3 × 4 works; 12 = 2 × 6 does not, because 2 and 6 share a factor.
- “NOT divisible by” questions — read the word not. Three options will pass.
Remainders
One equation does almost all of it.
Every remainder question is this one line rearranged. If n leaves 3 on division by 8, then n = 8k + 3 — write that down and the rest is algebra. The remainder is always smaller than the divisor; if yours is not, divide again.
When the integer n is divided by 8, the remainder is 3. What is the remainder if 14n is divided by 8? (SI 18 Aug 2021)
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Same start: n = 8k + 3, so 14n = 112k + 42. 112k is a multiple of 8 (112 = 8 × 14), so the remainder comes from 42: 42 = 8 × 5 + 2.
“Divided by 39 the remainder is 18 — what is the remainder on dividing by 13?” Since 13 divides 39, write n = 39k + 18. The 39k part vanishes, so you only divide 18 by 13: remainder 5. If the remainder is already smaller than the new divisor it does not change — that is why 483 → 23 with remainder 19 stays 19.
On dividing 29758 by a number, 234 is obtained as the quotient and 40 as the remainder. Find the divisor. (SI 15 Jul 2023)
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Dividend = divisor × quotient + remainder, so 29758 = d × 234 + 40. Then 234d = 29718 and d = 29718 ÷ 234 = 127.
What least number should be added to 1560 so that the resulting number is EXACTLY divisible by 29? (SI 18 Aug 2021)
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Divide 1560 by 29: 29 × 53 = 1537, so the remainder is 23. To reach the next multiple you need 29 − 23 = 6. (“Least number to subtract” would be the remainder itself, 23 — read which one is asked.)
HCF and LCM: 28 questions
The biggest single block in the chapter.
True for two numbers only, never for three. It turns three of the SI's questions into one division. It also settles the trick question: a valid pair must have HCF dividing LCM — 18 and 112 cannot be an HCF and LCM, because 18 does not divide 112.
Five bells start to toll together at intervals of 30, 36, 40, 48, and 60 minutes. If they start to toll at 9 : 00 a.m., at what time will they toll together again? (SI 14 Jul 2023)
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LCM of 30, 36, 40, 48, 60. In primes: 30 = 2·3·5, 36 = 2²·3², 40 = 2³·5, 48 = 2⁴·3, 60 = 2²·3·5. Take the highest power of each: 2⁴ × 3² × 5 = 16 × 9 × 5 = 720 minutes = 12 hours. 9:00 a.m. + 12 hours = 9:00 p.m.
The least number of square tiles required to pave a walking track in a park, which is 6 m 27 cm long and 11 m 97 cm broad, is : (SI 29 Jun 2023)
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Lengths in cm: 627 and 1197. The largest square that fits both exactly has side HCF(627, 1197). 627 = 3 × 11 × 19 and 1197 = 3² × 7 × 19, so the HCF is 3 × 19 = 57. Tiles = (627/57) × (1197/57) = 11 × 21 = 231.
Same remainder r for all divisors → the number is LCM + r. Smallest number leaving 1 with 5, 6, 8, 9, 15 is LCM(5,6,8,9,15) + 1 = 360 + 1 = 361, and the family is 360k + 1. Different remainders, but the same gap → subtract the gap. “Greatest number dividing 640, 710, 339 leaving 11, 7, 6” means it divides 629, 703 and 333 exactly, so the answer is their HCF, 37.
Find the smallest number which, when divided by 5, 6, 8, 9 and 15, leaves 1 as remainder in each case, but when divided by 13 leaves no remainder. (SI 15 Jul 2023)
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LCM(5, 6, 8, 9, 15) = 360, so the numbers leaving remainder 1 are 360k + 1: 361, 721, 1081, 1441, 1801, 2161, 2521, 2881, 3241, 3601... Now test which is divisible by 13. 3601 = 13 × 277, so the answer is 3601.
LCM of fractions = LCM of numerators ÷ HCF of denominators. HCF of fractions = HCF of numerators ÷ LCM of denominators. Reduce every fraction to lowest terms first. For 2/3, 4/5, 6/7: LCM(2,4,6) = 12 and HCF(3,5,7) = 1, so the LCM is 12. The SI has asked this three times.
- HCF × LCM = a × b holds only for two numbers. For three it is false.
- “Smallest 5-digit number divisible by …” — find the LCM, then the first multiple at or above 10000, not the LCM itself.
- Convert all lengths to the same unit before taking an HCF. 4 m 65 cm is 465 cm.
- A question asking for the number of tiles wants area ÷ area, not the HCF itself.
Fractions, decimals and number facts
23 questions, mostly one-liners.
Cross-multiply two at a time. For a/b and c/d, compare a×d with c×b — the bigger product sits above the bigger fraction. A faster check for fractions just below 1: the one whose numerator and denominator differ by less is the larger. 3/4, 5/6, 11/15, 7/8 differ by 1, 1, 4 and 1, so 11/15 is clearly the odd one out and the smallest.
Put the repeating digits over as many 9s, and shift for any non-repeating part. 3.4̅ (the 4 recurring) = 3 + 4/9 = 31/9. For 7.58333… only the 3 repeats: 7.58333… = 7 + 58̅/99… is messy, so use the standard move — multiply by 10 and by 100 and subtract. The SI's answer is 91/12, and 91 ÷ 12 = 7.58333… is the check that takes five seconds.
A certain amount is distributed among A, B and C. A gets 4/5 and B gets 1/6 of the whole amount. If C gets ₹15, then A gets : (SI 18 Aug 2021)
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A gets 4/5, B gets 1/6, so C gets 1 − 4/5 − 1/6 = (30 − 24 − 5)/30 = 1/30 of the total. C's 1/30 is ₹15, so the whole amount is ₹450 and A's share is 4/5 × 450 = ₹360.
- A number is prime if it has exactly two factors. 1 is neither prime nor composite.
- Between two consecutive squares n² and (n+1)² there are exactly 2n numbers. Between 25 and 36: 2 × 5 = 10.
- Three consecutive multiples of 9 summing to 2457: the middle one is 2457 ÷ 3 = 819, so they are 810, 819, 828.
- The smallest perfect square divisible by several numbers: take the LCM, then raise every prime to an even power. LCM(12,18,20) = 180 = 2²·3²·5, so multiply by 5 → 900.
Also asked onceTap to open
The remainder when 2^{64} is divided by 255 is : (SI 16 Oct 2022)
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2⁶⁴ ÷ 255. Note 255 = 2⁸ − 1, so 2⁸ leaves remainder 1 on division by 255. 2⁶⁴ = (2⁸)⁸, and 1⁸ = 1. Powers of 2 against 2ⁿ − 1 always collapse like this.
The difference between two numbers is 1240. On dividing the larger number by the smaller number, 4 is obtained as the quotient and 7 is left as the remainder. Find the larger of them. (SI 20 Jun 2023)
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Larger = 4 × smaller + 7 and larger − smaller = 1240. Substituting: 4s + 7 − s = 1240 ⇒ 3s = 1233 ⇒ s = 411. Larger = 411 + 1240 = 1651.
In an office, all the employees deposit as many paise as the number of employees. If the total amount collected is ₹81, then how many employees are there in the office? (SI 24 Aug 2021)
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Each employee deposits as many paise as there are employees, so n employees give n² paise. ₹81 = 8100 paise, so n² = 8100 and n = 90.
Find the sum of the digits of 2^{100} × 5^{104} × 10^{100}. (SI 20 Jun 2023)
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2¹⁰⁰ × 5¹⁰⁴ × 10¹⁰⁰ = 2¹⁰⁰ × 5¹⁰⁰ × 5⁴ × 10¹⁰⁰ = 10¹⁰⁰ × 625 × 10¹⁰⁰ = 625 followed by 200 zeros. The digit sum is 6 + 2 + 5 = 13.
In a toll-gate booth, the tariff of all four wheelers for every odd date is ₹100 and for even dates is ₹200. If the man paid a total of ₹3000 in all, for how many days did he pass through that toll-gate booth, given that the first day is the 5th date of the month? (SI 18 Aug 2021)
Show answer
A month has 15 or 16 odd dates and 15 or 14 even ones, but the clean reading is: let there be a odd-date visits and b even-date visits, 100a + 200b = 3000, so a + 2b = 30. The paper's answer is 20 visits, i.e. a = 10 and b = 10.
One-page revision
Read only this the night before.
- Divisible by 4?
- Last two digits form a multiple of 4
- Divisible by 8?
- Last three digits form a multiple of 8
- Divisible by 11?
- Odd-place sum − even-place sum = 0 or ±11
- HCF × LCM
- = product of the two numbers (two only)
- LCM of fractions
- LCM of numerators ÷ HCF of denominators
- HCF of fractions
- HCF of numerators ÷ LCM of denominators
- Same remainder r for all divisors
- LCM × k + r
- Divisors leaving different remainders
- Subtract each remainder, then take the HCF
- Numbers between n² and (n+1)²
- 2n
- Sum and difference given
- Larger = (sum + difference)/2
Two sentences cover most of this chapter: a number leaving remainder r on division by d is d×k + r, and HCF is for sharing out, LCM is for coming back together.
Every remaining question, by tool
Forty-nine more, each answered by one of the four tools above.
Divisibility and remaindersTap to open
Which of the following numbers is divisible by 15? (SI 9 Jul 2023)
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Divisible by 15 means divisible by both 3 and 5. Only 47325 and 23755 end in 0 or 5; of those, 4+7+3+2+5 = 21 is a multiple of 3 while 2+3+7+5+5 = 22 is not. So the answer is 47325.
Which of the following numbers is not divisible by 4? (SI 16 Oct 2022)
Show answer
A number is divisible by 4 when its last two digits are. Check them: 44, 64, 22 and 92. Of these only 22 is not a multiple of 4. So 188822 is the one that is not divisible by 4.
Which of these is completely divisible by 12? (SI 20 Jun 2023)
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Divisible by 12 means divisible by 4 and 3. Last two digits: 96, 56, 45, 41 — only 96 and 56 pass the 4 test. Digit sums: 45628796 gives 47 (not a multiple of 3); 45236856 gives 39, which is. So the answer is 45236856.
56352744 is completely divisible by x. What is the value of x? (SI 2 Jul 2023)
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Test the options with the quick rules. 56352744 ends in 44, so it passes the 4 test; its digit sum is 36, so it passes the 3 test; and 744 ÷ 8 = 93, so it passes the 8 test. Divisible by 8 and by 3 means divisible by 24. It fails 11 (alternating sum 4 − 4 + 7 − 2 + 5 − 3 + 6 − 5 = 8) and 13, so 24 is the only option that works.
In the following sequence of numbers identify the number which is divisible by 11 : 836, 1156, 4624, 26578 (SI 18 Aug 2021)
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For 11, take the alternating sum of the digits. For 836: 8 − 3 + 6 = 11, a multiple of 11. The others give 1156 → 1 − 1 + 5 − 6 = −1; 4624 → 4 − 6 + 2 − 4 = −4; 26578 → 2 − 6 + 5 − 7 + 8 = 2. So the answer is 836.
Which of the following is the smallest number of 4 digits that is completely divisible by 11? (SI 14 Jul 2023)
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Among the options only 1001, 9900, 1232 and 1210 are all multiples of 11 — but the question asks for the smallest four-digit one. 1001 is the smallest, and indeed 1001 = 11 × 91. So the answer is 1001.
The number 6×30+135×14+105×12 is divisible by ______. (SI 14 Jul 2023)
Show answer
Evaluate first: 6 × 30 = 180, 135 × 14 = 1,890 and 105 × 12 = 1,260. The total is 3,330. Now 3,330 = 2 × 3² × 5 × 37, so of the options only 37 divides it.
Which of the following numbers is exactly divisible by 101, 103 and 105? (SI 19 Aug 2021)
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A number divisible by 101, 103 and 105 must be a multiple of their product, 101 × 103 × 105 = 1,092,315. That is exactly option B, 1092315. The three numbers share no common factor, so their LCM is simply the product.
If 46a6890432b is divisible by both 9 and 5, then which of the following is the greatest possible value of ab? (SI 19 Aug 2021)
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Divisible by 5 means the last digit b is 0 or 5. Divisible by 9 means the digit sum 4+6+a+6+8+9+0+4+3+2+b = 42 + a + b is a multiple of 9. If b = 0, then 42 + a must be a multiple of 9, so a = 3 and ab = 0. If b = 5, then 47 + a must be, so a = 7 and ab = 35. The greatest possible value of ab is 35.
What is the remainder when 8127 is divided by 8? (SI 2 Jul 2023)
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8127 = 8 × 1015 + 7, so the remainder is 7. A faster route: 8000 and 120 are both multiples of 8, leaving 7.
If a positive integer n is divided by 8, the remainder is 5. Which of the numbers below yields a remainder of 0 when it is divided by 8? (SI 9 Jul 2023)
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n = 8k + 5, so adding 3 gives 8k + 8 = 8(k + 1), a clean multiple of 8. The answer is n + 3. In general, to clear a remainder r when dividing by d, add d − r.
When a number is divided by 483, the remainder is 19. What will be the remainder when the same number is divided by 23? (SI 31 Jul 2023)
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483 = 21 × 23, so a multiple of 483 is also a multiple of 23. The number is 483k + 19, and the 483k part leaves nothing behind when divided by 23. Since 19 is already smaller than 23, the remainder is still 19.
Find the largest 4-digit number which, when divided by 2, 5, 7 or 10, leaves the same remainder 1 in each case. (SI 9 Jul 2023)
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“Same remainder 1 when divided by 2, 5, 7 or 10” means the number is 1 more than a common multiple. LCM(2, 5, 7, 10) = 70. The largest 4-digit multiple of 70 is 9,940 (since 9,999 ÷ 70 = 142 remainder 59, and 142 × 70 = 9,940). Adding 1 gives 9941.
The smallest multiple of 7 which, when divided by 9, 18, 6 and 15, leaves a remainder of 4 : (SI 15 Jul 2023)
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The number must be 4 more than a common multiple of 9, 18, 6 and 15, whose LCM is 90. So it has the form 90k + 4. Testing k = 1, 2, 3, 4: 94, 184, 274, 364 — and 364 = 7 × 52 is the first that is also a multiple of 7. The answer is 364.
HCF and LCMTap to open
The LCM of 2,3 and 4 is : (SI 29 Jul 2023)
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LCM(2, 3, 4) = 12.
Find the smallest number that is exactly divisible by 15, 24 and 30. (SI 29 Jun 2023)
Show answer
15 = 3 × 5, 24 = 2³ × 3 and 30 = 2 × 3 × 5. Taking the highest power of each prime gives 2³ × 3 × 5 = 120.
Find the LCM of 7, 12, 15 and 60. (SI 20 Jun 2023)
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60 is already a multiple of 12 and 15, so only 7 is new: LCM = 60 × 7 = 420.
Find the lowest common multiple (L.C.M) of 34, 45, and 50. (SI 16 Oct 2022)
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34 = 2 × 17, 45 = 3² × 5 and 50 = 2 × 5². The LCM takes 2 × 3² × 5² × 17 = 2 × 9 × 25 × 17 = 7650.
The least number that is divisible by 675 and 550 is equal to : (SI 14 Jul 2023)
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“The least number divisible by both” is the LCM. 675 = 3³ × 5² and 550 = 2 × 5² × 11, so the LCM is 2 × 3³ × 5² × 11 = 14850. Option A (270) is a common factor trap — 270 is smaller than 675, so it cannot be a multiple of it.
The HCF and LCM of two numbers are 4 and 1008, respectively. If one of the numbers is 112, then the other is : (SI 14 Jul 2023)
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For any two numbers, HCF × LCM = the product of the numbers. So the other number is 4 × 1008 ÷ 112 = 4032/112 = 36.
If the LCM and HCF of two numbers are 72 and 6, respectively. If one number is 18, then the second number is (SI 29 Jul 2023)
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Same rule: the second number is 72 × 6 ÷ 18 = 24.
Three numbers are in the ratio 3 : 4 : 5 and their LCM is 2400. Then, their HCF is : (SI 29 Jul 2023)
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Write the numbers as 3x, 4x and 5x, where x is the HCF (3, 4 and 5 share no factor). Their LCM is LCM(3, 4, 5) × x = 60x = 2,400, so x = 40.
The LCM of 3/5, 4/5 and 6/11 will be : (SI 9 Jul 2023)
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The LCM of fractions is LCM of the numerators ÷ HCF of the denominators. Numerators 3, 4, 6 have LCM 12; denominators 5, 5, 11 have HCF 1. So the answer is 12/1 = 12.
The LCM of 2/3, 4/5 and 6/7 is : (SI 15 Jul 2023)
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Same rule: LCM(2, 4, 6) = 12 and HCF(3, 5, 7) = 1, so the answer is 12.
What is the LCM of 2/3, 8/9, 16/81, 10/27? (SI 31 Jul 2023)
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Numerators 2, 8, 16, 10 have LCM 80. Denominators 3, 9, 81, 27 have HCF 3. So the LCM of the fractions is 80/3.
Bhim has 90 pencils and 216 pens. He wants to distribute the pencils and pens such that each student gets the same number of pens and pencils. What is the maximum number of students among whom he can distribute the pens and pencils? (SI 2 Jul 2023)
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Each student must get a whole number of both, so the number of students divides 90 and 216 — that is the HCF. 90 = 2 × 3² × 5 and 216 = 2³ × 3³, so HCF = 2 × 3² = 18. “Same number each, maximum people” always means HCF; “all together again, soonest” always means LCM.
Find the maximum length a scale can have if it can be used to measure the lengths 8 m, 4 m 65 cm and 13 m 65 cm exactly. (SI 20 Jun 2023)
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Convert to one unit: 8 m = 800 cm, 4 m 65 cm = 465 cm and 13 m 65 cm = 1,365 cm. The longest scale that measures all three exactly is their HCF. 465 = 3 × 5 × 31 and 800 = 2⁵ × 5², sharing only 5; 1,365 is also a multiple of 5. So the answer is 5 cm.
Four different sweet varieties of count 64, 288, 96 and 168 were ordered for a particular occasion. They need to be packed in such a way that each box has the same variety of sweet and the number of sweets in each box is also the same. What is the minimum number of boxes required to pack? (SI 18 Aug 2021)
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Each box holds one variety and the same count, so the count per box is the HCF of 64, 288, 96 and 168, which is 8. The number of boxes is then 64/8 + 288/8 + 96/8 + 168/8 = 8 + 36 + 12 + 21 = 77. Using the HCF makes each box as full as possible, which is what minimises the box count.
What is the minimum number of square sheets required for a floor of length 5 m 78 cm and breadth 3 m 74 cm? (SI 24 Aug 2021)
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Square sheets of the largest possible side use the HCF of 578 cm and 374 cm. 578 = 2 × 17² and 374 = 2 × 11 × 17, so the HCF is 2 × 17 = 34 cm. The floor then takes (578/34) × (374/34) = 17 × 11 = 187 sheets.
The least number which should be added to 2014 so that the sum is exactly divisible by 3, 4, 5 and 6 is : (SI 16 Oct 2022)
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LCM(3, 4, 5, 6) = 60. Dividing 2,014 by 60 leaves a remainder of 34. To reach the next multiple of 60 you must add 60 − 34 = 26.
The smallest 5-digit number which is exactly divisible by 9, 14 and 36 is : (SI 18 Aug 2021)
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LCM(9, 14, 36) = 252. The smallest 5-digit number is 10,000, and 10,000 ÷ 252 = 39 remainder 172. So the next multiple is 40 × 252 = 10080.
Find the greatest possible five-digit number that is completely divisible by 3, 6 and 9. (SI 29 Jun 2023)
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Divisible by 3, 6 and 9 means divisible by their LCM, which is 18 — note it must be even, because of the 6. 99,999 is odd, so it fails. 99,999 ÷ 18 = 5,555 remainder 9, so the largest multiple is 5,555 × 18 = 99990.
How many numbers between 31 and 250 are divisible by 2, 3 and 5? (SI 20 Jun 2023)
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Divisible by 2, 3 and 5 means divisible by 30. Between 31 and 250 the multiples of 30 are 60, 90, 120, 150, 180, 210 and 240 — 7 numbers.
The least number, which when divided by 108, 171 and 261, leaves the remainder 5 in each case is : (SI 29 Jun 2023)
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The number is 5 more than a common multiple of 108, 171 and 261. 108 = 2² × 3³, 171 = 3² × 19 and 261 = 3² × 29, so the LCM is 2² × 3³ × 19 × 29 = 59,508. Adding 5 gives 59513.
What is the largest 4-digit number which is added to 5349 so that the resulting number when divided by 15, 18, 32, 48 and 64 leaves a remainder of 7 in each case? (SI 29 Jun 2023)
Show answer
LCM(15, 18, 32, 48, 64) = 2,880, so 5,349 + N must be 2,880k + 7. That means N = 2,880k + 7 − 5,349 = 2,880k − 5,342. For k = 5 this is 14,400 − 5,342 = 9,058, which is a 4-digit number; k = 6 would give 11,938, which has five digits. So the largest 4-digit value is 9058.
Fractions and decimalsTap to open
Which of the following fractions is the smallest? (SI 16 Oct 2022)
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Cross-compare rather than finding a common denominator for all four. 14/17 ≈ 0.82, 11/17 ≈ 0.65, 13/21 ≈ 0.62 and 5/9 ≈ 0.56. The smallest is 5/9.
Which of the following sets of fractions are in descending order? (SI 16 Oct 2022)
Show answer
Convert to decimals: 11/3 ≈ 3.67, 6/7 ≈ 0.857, 5/7 ≈ 0.714 and 3/5 = 0.6. Descending means largest first, so the correct order is 11/3, 6/7, 5/7, 3/5 — option C.
A certain amount of ₹25,760 is distributed among A, B and C. A gets 2/5 and B gets 1/5 of the whole amount. How much will C get? (SI 18 Aug 2021)
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A takes 2/5 and B takes 1/5, so C takes 1 − 3/5 = 2/5 of the total. That is 2/5 × 25,760 = ₹10,304.
When 0.26 is written in simplest fractional form, the sum of numerator and the denominator is _________________. (SI 18 Aug 2021)
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0.26 = 26/100, and dividing both by 2 gives 13/50 in simplest form. The sum of numerator and denominator is 13 + 50 = 63.
If 0.45 is written in its simplest fractional form, what will be the difference between its numerator and denominator? (SI 29 Jun 2023)
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0.45 = 45/100 = 9/20 in simplest form. The question asks numerator minus denominator, so 9 − 20 = −11. The minus sign is the point of the question.
What is the fraction equal to 7.5833... (the digit 3 recurring)? (SI 9 Jul 2023)
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Use the recurring-decimal rule: for 7.58̅3̅, subtract the non-recurring part. Write x = 7.58333…, then 1000x = 7583.333… and 100x = 758.333… Subtracting, 900x = 6825, so x = 6825/900 = 91/12. A faster check: 0.58333… = 7/12, so the number is 7 + 7/12 = 91/12 ≈ 7.583.
Which of the following is a recurring decimal? (SI 29 Jul 2023)
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A fraction gives a terminating decimal exactly when the denominator's only prime factors are 2 and 5. Here 2, 10 and 5 all qualify, but 3 does not. So 1/3 = 0.333… is the recurring one.
Number propertiesTap to open
How many natural numbers are there between 5² and 6², not including either of these perfect squares? (SI 18 Aug 2021)
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5² = 25 and 6² = 36. The natural numbers strictly between them are 26 through 35, which is 35 − 26 + 1 = 10 numbers. In general there are 2n numbers between n² and (n + 1)², and here 2 × 5 = 10.
Which of the following is the largest number of 4 digits that is completely divisible by 16? (SI 29 Jun 2023)
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9,999 ÷ 16 = 624 remainder 15, so the largest 4-digit multiple of 16 is 624 × 16 = 9984.
The sum of three consecutive multiples of 9 is 2457. Find the largest number. (SI 2 Jul 2023)
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Three consecutive multiples of 9 are 9n − 9, 9n and 9n + 9, and their sum is 27n. So 27n = 2,457 and n = 91, making the middle number 819 and the largest 819 + 9 = 828. Option A (819) is the middle one, placed there for anyone who stops early.
Three times a number is divisible by 12 and two times the number is divisible by 18. The number is divisible by _________. (SI 15 Jul 2023)
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3n divisible by 12 means n is divisible by 4. 2n divisible by 18 means n is divisible by 9. A number divisible by both 4 and 9 is divisible by 36.
Find the least square number which is exactly divisible by 4, 9 and 15. (SI 16 Oct 2022)
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LCM(4, 9, 15) = 180 = 2² × 3² × 5. For a perfect square every prime must appear an even number of times, and only the 5 falls short. Multiplying by 5 gives 900 = 30², so the answer is 900.
How many 3 digit numbers can be formed using the numbers 6, 1, 2, 3 without repetition ? (SI 2016)
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Four digits with no repetition: 4 choices for the first place, 3 for the second and 2 for the third, giving 4 × 3 × 2 = 24.
If all the digits in the number '6231732548' are arranged in ascending order, then how many digits remain in the same place? (SI 16 Oct 2022)
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Write the digits out and sort them. Original: 6 2 3 1 7 3 2 5 4 8 Ascending: 1 2 2 3 3 4 5 6 7 8 Comparing position by position, only the 2nd digit (2) and the 3rd digit (3) stay put, so the answer is two.