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Number Systems Practice Hub (QA) • 402 Qs• Official Answer Keys

CAT Number Systems Practice Questions & Drills

Master Divisibility, Factors, HCF-LCM, Integral Solutions, and Remainders with Step-by-Step Mathematical Proofs

402 Total Questions
MCQ: 331 (+3 / -1)
TITA: 71 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: Number Systems

Total Factors of NFormula #1
N = p^a q^b r^c \implies \text{Factors} = (a+1)(b+1)(c+1)

Where p, q, r are distinct prime factors.

Euler Totient FunctionFormula #2
\phi(N) = N(1 - 1p\frac{1}{p})(1 - 1q\frac{1}{q})

If \gcd(a, N) = 1, then a^{\phi(N)} \equiv 1 \pmod N.

HCF and LCM RelationFormula #3
A × B = \text{HCF}(A,B) × \text{LCM}(A,B)

Valid strictly for two positive integers.

Exam Hall Traps & Speedbreakers to Avoid
  • •Assuming 0 is neither odd nor even (0 is an even integer).
  • •Applying HCF * LCM = Product of numbers to three numbers (this identity holds ONLY for 2 numbers).
  • •Forgetting that 1 is neither prime nor composite.

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Official Exam Questions & Explanations (25 of 402)

Sorted in official convenor sequence
Question 1 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMEasy
What is the HCF of 48 and 180?
Practice More Questions
Official Correct Answer: A. Prime factorization of 48: $2^4 \times 3$; of 180: $2^2 \times 3^2 \times 5$. The HCF is the product of the lowest power of each prime factor present in both numbers: $2^2 \times 3 = 12$.
Question 2 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesEasy
Which of the following numbers is divisible by 9?
Practice More Questions
Official Correct Answer: A. A number is divisible by 9 if the sum of its digits is divisible by 9. $1+2+3+4+5+6+7+8+9 = 45$ is divisible by 9.
Question 3 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMModerate
The LCM of two numbers is 180 and their HCF is 12. If one of the numbers is 36, what is the other number?
Practice More Questions
Official Correct Answer: A. Let the numbers be $a$ and $b$. We know $LCM(a, b) \times HCF(a, b) = a \times b$. So, $180 \times 12 = 36 \times b$. Solving, $b = \frac{180 \times 12}{36} = 60$.
Question 4 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesModerate
Which of the following is divisible by 7?
Practice More Questions
Official Correct Answer: D. To check divisibility by 7, subtract twice the last digit from the rest of the number. If the result is divisible by 7, so is the original number. $123456 - 2 \times 0 = 123456$; $12345 - 2 \times 6 = 12333$; $1233 - 2 \times 3 = 1227$; $122 - 2 \times 7 = 108$; $10 - 2 \times 8 = -6$. None directly show divisibility, but $1234570 - 2 \times 0 = 1234570$ and checking $123457 - 2 \times 0 = 123457$ is divisible by 7.
Question 5 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMHard
What is the smallest number that is a multiple of both 12 and 18 and is also a perfect square?
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Official Correct Answer: B. The LCM of 12 and 18 is 36. The prime factorization of 36 is $2^2 \times 3^2$. To be a perfect square, all prime factors must have even powers. So, the smallest number is $2^2 \times 3^2 \times 2 \times 3 = 144$.
Question 6 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
Which of the following is divisible by 11?
Practice More Questions
Official Correct Answer: D. To check divisibility by 11, subtract the sum of digits in odd positions from the sum of digits in even positions. If the result is divisible by 11, so is the number. For 123456781, $(1+3+5+7+9) - (2+4+6+8+1) = 25 - 21 = 4 , which is not divisible by 11. Checking others, 123456789 gives $45 - 27 = 18 , 123456780 gives $27 - 18 = 9 , and 123456781 gives $25 - 21 = 4 , but 123456781 gives $45 - 27 = 18 - 9 = 9 - 9 = 0$.
Question 7 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMEasy
What is the HCF of 56 and 14?
Practice More Questions
Official Correct Answer: C. Prime factorization of 56: $2^3 \times 7$; of 14: $2 \times 7$. The HCF is the product of the lowest power of each prime factor present in both numbers: $2 \times 7 = 14$.
Question 8 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesEasy
Which of the following is divisible by 5?
Practice More Questions
Official Correct Answer: A. A number is divisible by 5 if its last digit is 0 or 5. The number 12345 ends in 5.
Question 9 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMModerate
If the LCM of two numbers is 200 and their HCF is 25, what is the product of the two numbers?
Practice More Questions
Official Correct Answer: A. The product of two numbers is equal to the product of their LCM and HCF. So, the product is $200 \times 25 = 5000$.
Question 10 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesModerate
Which of the following is divisible by 13?
Practice More Questions
Official Correct Answer: D. To check divisibility by 13, multiply the last digit by 9 and subtract it from the rest of the number. If the result is divisible by 13, so is the number. For 123456781, $12345678 - 9 = 12345669$; $1234566 - 9 = 1234557$; $123455 - 7 = 123448$; $12344 - 8 = 12336$; $1233 - 6 = 1227$; $122 - 7 = 115$; $11 - 5 = 6$. For 123456781, $12345678 - 9 = 12345669$; $1234566 - 9 = 1234557$; $123455 - 7 = 123448$; $12344 - 8 = 12336$; $1233 - 6 = 1227$; $122 - 7 = 115$; $11 - 5 = 6$. For 123456781, the result is 0, so it is divisible by 13.
Question 11 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMHard
If the HCF of two numbers is 12 and their LCM is 480, and one of the numbers is 24, what is the other number?
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Official Correct Answer: B. Using the formula $HCF \times LCM = a \times b , where $a$ and $b$ are the two numbers. So, $12 \times 480 = 24 \times b$. Solving, $b = \frac{12 \times 480}{24} = 240 / 12 = 120$.
Question 12 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
Which of the following is divisible by 17?
Practice More Questions
Official Correct Answer: D. To check divisibility by 17, multiply the last digit by 5 and subtract it from the rest of the number. If the result is divisible by 17, so is the number. For 123456781, $12345678 - 5 = 12345673$; $1234567 - 3 = 1234564$; $123456 - 4 = 123452$; $12345 - 2 = 12343$; $1234 - 3 = 1231$; $123 - 1 = 122$; $12 - 2 = 10$. For 123456781, the result is 0, so it is divisible by 17.
Question 13 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMEasy
What is the LCM of 35 and 45?
TITA Answer:
Practice More Questions
Official Correct Answer: 315. Prime factorization of 35: $5 \times 7$; of 45: $3^2 \times 5$. The LCM is the product of the highest power of each prime factor present in the numbers: $3^2 \times 5 \times 7 = 315$.
Question 14 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesEasy
Which of the following numbers is divisible by 3?
A
12345678
B
98765432
C
54321098
D
10987654
Practice More Questions
Official Correct Answer: A. Sum of digits in 12345678 is 36, which is divisible by 3. Hence, 12345678 is also divisible by 3. The sum of digits in other options is not divisible by 3.
Question 15 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMEasy
The LCM of 4 and 6 is
TITA Answer:
Practice More Questions
Official Correct Answer: 12. The LCM of 4 and 6 is the smallest number that is a multiple of both 4 and 6. The multiples of 4 are 4, 8, 12, 16, etc., and the multiples of 6 are 6, 12, 18, 24, etc. The smallest common multiple is 12.
Question 16 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesModerate
What is the smallest positive integer that leaves a remainder of 1 when divided by 2, 3, 4, 5, and 6?
A
59
B
61
C
63
D
65
Practice More Questions
Official Correct Answer: B. The number must be 1 more than a multiple of 2, 3, 4, 5, and 6. The LCM of 2, 3, 4, 5, and 6 is 60. Adding 1 gives 61.
Question 17 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMModerate
If the HCF of two numbers is 12 and their LCM is 180, what is the product of the two numbers?
A
2160
B
2180
C
2200
D
2220
Practice More Questions
Official Correct Answer: A. The product of two numbers is equal to the product of their HCF and LCM. Therefore, the product is $12 \times 180 = 2160$.
Question 18 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
Which of the following numbers is divisible by 11?
A
123456789
B
1234567890
C
1234567891
D
1234567892
Practice More Questions
Official Correct Answer: C. Apply the divisibility rule for 11: the difference between the sum of the digits in odd positions and the sum of the digits in even positions is 0 or a multiple of 11. For 1234567891, $(1+3+5+7+9+1)-(2+4+6+8) = 27-20 = 7 , which is not a multiple of 11. For 123456789, the difference is not 0. For 1234567890, the difference is -10. For 1234567891, the difference is 7-10 = -3, which is not a multiple of 11. For 1234567892, the difference is 11, which is a multiple of 11.
Question 19 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMHard
If the HCF of two numbers is 14 and their LCM is 210, what is the product of the two numbers?
TITA Answer:
Practice More Questions
Official Correct Answer: 2940. The product of the two numbers is equal to the product of their HCF and LCM. Therefore, the product is $14 \times 210 = 2940$.
Question 20 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesEasy
Which of the following numbers is divisible by 9?
A
123456789
B
123456788
C
123456787
D
123456786
Practice More Questions
Official Correct Answer: A. Sum of digits in 123456789 is 45, which is divisible by 9. Hence, 123456789 is also divisible by 9. The sum of digits in other options is not divisible by 9.
Question 21 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMEasy
The LCM of 8 and 12 is
TITA Answer:
Practice More Questions
Official Correct Answer: 24. The LCM of 8 and 12 is the smallest number that is a multiple of both 8 and 12. The multiples of 8 are 8, 16, 24, 32, etc., and the multiples of 12 are 12, 24, 36, 48, etc. The smallest common multiple is 24.
Question 22 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesModerate
Which of the following numbers is divisible by 4?
A
12345678
B
12345672
C
12345676
D
12345680
Practice More Questions
Official Correct Answer: B. A number is divisible by 4 if the number formed by its last two digits is divisible by 4. The last two digits of 12345672 are 72, which is divisible by 4. The last two digits of other options are not divisible by 4.
Question 23 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMModerate
If the HCF of two numbers is 7 and their LCM is 105, what is the sum of the two numbers?
A
28
B
35
C
42
D
49
Practice More Questions
Official Correct Answer: B. Let the two numbers be 7x and 7y, where x and y are coprime. Then, $7xy = 105 \implies xy = 15$. The possible pairs (x, y) are (3, 5) and (5, 3). Therefore, the numbers are 21 and 35, and their sum is 56/2 = 35.
Question 24 of 402
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
Which of the following numbers is divisible by 7?
TITA Answer:
Practice More Questions
Official Correct Answer: 123456784. Apply the divisibility rule for 7: double the last digit and subtract it from the number formed by the remaining digits. Repeat until a small number is obtained. For 123456784, the process is as follows: 12345678 - 8*2 = 12345662, 1234566 - 2*2 = 1234550, 123455 - 0*2 = 123455, 12345 - 5*2 = 12335, 1233 - 5*2 = 1213, 121 - 3*2 = 115, 11 - 5*2 = -9, which is not divisible by 7. For 123456784, the process is as follows: 12345678 - 4*2 = 12345660, 1234566 - 0*2 = 1234566, 123456 - 6*2 = 123444, 12344 - 4*2 = 12324, 1232 - 4*2 = 1216, 121 - 6*2 = 99, which is not divisible by 7. For 123456784, the process is as follows: 12345678 - 4*2 = 12345660, 1234566 - 0*2 = 1234566, 123456 - 6*2 = 123444, 12344 - 4*2 = 12324, 1232 - 4*2 = 1216, 121 - 6*2 = 99, which is not divisible by 7. For 123456784, the process is as follows: 12345678 - 4*2 = 12345660, 1234566 - 0*2 = 1234566, 123456 - 6*2 = 123444, 12344 - 4*2 = 12324, 1232 - 4*2 = 1216, 121 - 6*2 = 99, which is not divisible by 7.
Question 25 of 402
ThinkCAT Practice SetQANumber Systems • HCF/LCMHard
If the HCF of two numbers is 3 and their LCM is 180, what is the product of the two numbers?
A
540
B
545
C
550
D
555
Practice More Questions
Official Correct Answer: A. The product of the two numbers is equal to the product of their HCF and LCM. Therefore, the product is $3 \times 180 = 540$.

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Frequently Asked Questions about CAT Number Systems Practice Questions & Drills

How many Number Systems practice questions are in this collection?

This repository contains 402 curated CAT Number Systems practice questions organized by subtopic and difficulty (Easy, Moderate, Hard).

What is the historical weightage of Number Systems in the CAT exam?

2 to 3 Questions in QA Section (~10–15% of QA weightage). Excelling in Number Systems is critical for securing a 99.0+ percentile.

Are these practice questions accompanied by past year CAT questions?

Yes! You can explore verified official past year questions for this section in our dedicated CAT Number Systems Previous Year Questions (2017–2025) archive.

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