24 questions, 14 of them short answer (type the number, as in IPMAT Indore). Each is marked with the exam level it suits. Answer, then open the step-by-step solution.
A number leaves remainders 3, 5 and 8 when divided by 4, 6 and 9. What is the smallest such number greater than 36?
Each remainder is one less than the divisor, so n + 1 is a multiple of each.
LCM = 36, so n + 1 is a multiple of 36.
The required number is 71.
Find the remainder when the 57-digit number 444…444 (the digit 4 written 57 times) is divided by 11.
Key fact: 11 divides 11, so pairs of digits drop out.
Removing the blocks that are multiples of 11 leaves a short number with the same remainder.
Remainder = 4.
What is the largest number that divides 199, 316 and 537 leaving the same remainder in each case?
If each leaves remainder r, the divisor divides every difference.
Differences: 117 and 221; their HCF is 13.
So the answer is 13 (the common remainder is 4).
Find the remainder when 27210 is divided by 5.
27 ≡ 2 (mod 5).
24 ≡ 1 (mod 5), so powers repeat every 4.
210 = 4 × 52 + 2, so 27210 ≡ 22 ≡ 4 (mod 5).
Find the remainder when the 55-digit number 666…666 (the digit 6 written 55 times) is divided by 9.
Key fact: the digit sum decides the remainder.
Removing the blocks that are multiples of 9 leaves a short number with the same remainder.
Remainder = 6.
What is the largest number that divides 252, 275 and 1126 leaving the same remainder in each case?
If each leaves remainder r, the divisor divides every difference.
Differences: 23 and 851; their HCF is 23.
So the answer is 23 (the common remainder is 22).
How many integers from 1 to 120 have no common factor with 120 other than 1?
120 = 23 × 3 × 5.
Count = 120 × (1 − 12) × (1 − 13) × (1 − 15) = 32.
How many integers from 1 to 36 have no common factor with 36 other than 1?
36 = 22 × 32.
Count = 36 × (1 − 12) × (1 − 13) = 12.
Find the remainder when the 54-digit number 333…333 (the digit 3 written 54 times) is divided by 7.
Key fact: 111111 is divisible by 7.
Removing the blocks that are multiples of 7 leaves a short number with the same remainder.
Remainder = 0.
Find the remainder when the 66-digit number 666…666 (the digit 6 written 66 times) is divided by 11.
Key fact: 11 divides 11, so pairs of digits drop out.
Removing the blocks that are multiples of 11 leaves a short number with the same remainder.
Remainder = 0.
How many integers from 1 to 48 have no common factor with 48 other than 1?
48 = 24 × 3.
Count = 48 × (1 − 12) × (1 − 13) = 16.
How many integers from 1 to 60 have no common factor with 60 other than 1?
60 = 22 × 3 × 5.
Count = 60 × (1 − 12) × (1 − 13) × (1 − 15) = 16.
A total of 7189 digits is used to number the pages of a book, starting from page 1. How many pages does it have?
Pages 1–9 use 9 digits and 10–99 use 180 more (189 in all).
Pages 100–999 use 2700 more (2889 in all). The rest, 4300, are 4-digit pages: 1075.
Pages = 2074.
What is the smallest 4-digit number that leaves remainder 3 when divided by each of 4, 6 and 9?
LCM(4, 6, 9) = 36; the number has the form 36k + 3.
The smallest such number with 4 digits is 1011.
What is the smallest 3-digit number that leaves remainder 4 when divided by each of 6, 8 and 12?
LCM(6, 8, 12) = 24; the number has the form 24k + 4.
The smallest such number with 3 digits is 100.
Find the remainder when 21490 is divided by 11.
21 ≡ 10 (mod 11).
102 ≡ 1 (mod 11), so powers repeat every 2.
490 = 2 × 245 + 0, so 21490 ≡ 100 ≡ 1 (mod 11).
What is the largest number that divides 181, 1357 and 1381 leaving the same remainder in each case?
If each leaves remainder r, the divisor divides every difference.
Differences: 1176 and 24; their HCF is 24.
So the answer is 24 (the common remainder is 13).
What is the largest number that divides 328, 448 and 898 leaving the same remainder in each case?
If each leaves remainder r, the divisor divides every difference.
Differences: 120 and 450; their HCF is 30.
So the answer is 30 (the common remainder is 28).
Find the remainder when 3363 is divided by 7.
Work with powers of 3 modulo 7.
36 ≡ 1 (mod 7), so powers repeat every 6.
363 = 6 × 60 + 3, so 3363 ≡ 33 ≡ 6 (mod 7).
How many digits are used to number the pages of a book with 875 pages?
Pages 1–9 use 9 digits, 10–99 use 180 digits, 100–999 use 2700 digits, and so on.
Total for 875 pages = 2517.
A total of 2697 digits is used to number the pages of a book, starting from page 1. How many pages does it have?
Pages 1–9 use 9 digits and 10–99 use 180 more (189 in all).
The remaining 2508 digits are 3-digit pages: 836.
Pages = 935.
What is the smallest 3-digit number that leaves remainder 1 when divided by each of 7, 9 and 12?
LCM(7, 9, 12) = 252; the number has the form 252k + 1.
The smallest such number with 3 digits is 253.
How many digits are used to number the pages of a book with 967 pages?
Pages 1–9 use 9 digits, 10–99 use 180 digits, 100–999 use 2700 digits, and so on.
Total for 967 pages = 2793.
Find the remainder when 1082 is divided by 13.
Work with powers of 10 modulo 13.
106 ≡ 1 (mod 13), so powers repeat every 6.
82 = 6 × 13 + 4, so 1082 ≡ 104 ≡ 3 (mod 13).
These are original practice questions written by Target Campus in the pattern, syllabus and difficulty of IPMAT Indore, IPMAT Rohtak, JIPMAT, NPAT and SET. They are not previous-year papers.
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