ThinkCAT Logo
ThinkCAT
Number Systems • Divisibility Rules Practice Drills (110 Qs)• Official Answer Keys

CAT Divisibility Rules Practice Questions (110+ Questions)

Divisibility by 3, 7, 9, 11, 13, Composite Divisibility Checks, and Missing Digit Problems

110 Total Questions
MCQ: 94 (+3 / -1)
TITA: 16 (0 Negative Penalty)
Launch Divisibility Rules Practice Drills Drill in Study ModeInteractive Practice (5,200+ Qs)

Core Formulas & Shortcut Matrix: Divisibility Rules Practice Drills

Divisibility by 11Formula #1
\sum \text{Odd Place Digits} - \sum \text{Even Place Digits} \equiv 0 ±od{11}

Difference must be a multiple of 11 (including 0).

Divisibility by 7, 11, 13 (Group of 3s)Formula #2
\text{Alternating sum of 3-digit blocks} ±od{1001}

Since 7 * 11 * 13 = 1001.

Exam Hall Traps & Speedbreakers to Avoid
  • •Applying single divisibility rule to composite numbers without splitting into co-prime factors (e.g. 12 = 3 * 4, not 2 * 6).

Bite-Sized Practice Sets (11 Modules Available)

Authentic CAT Blueprint

Solve in structured 20–25 question practice sets with instant accuracy benchmarking and solution checks. Sets 01 & 02 are completely free.

12 QuestionsFree Pass

Set 01: Divisibility Rules

Difficulty: Easy
Start Free Set
6 QuestionsPro Pass

Set 02: Divisibility Rules

Difficulty: Hard
Unlock Set (Pro)
8 QuestionsPro Pass

Set 03: Divisibility Rules

Difficulty: Easy
Unlock Set (Pro)
14 QuestionsPro Pass

Set 04: Divisibility Rules

Difficulty: Easy
Unlock Set (Pro)
13 QuestionsPro Pass

Set 05: Divisibility Rules

Difficulty: Moderate
Unlock Set (Pro)
13 QuestionsPro Pass

Set 06: Divisibility Rules

Difficulty: Easy
Unlock Set (Pro)
12 QuestionsPro Pass

Set 07: Divisibility Rules

Difficulty: Hard
Unlock Set (Pro)
13 QuestionsPro Pass

Set 08: Divisibility Rules

Difficulty: Hard
Unlock Set (Pro)
7 QuestionsPro Pass

Set 09: Divisibility Rules

Difficulty: Easy
Unlock Set (Pro)
8 QuestionsPro Pass

Set 10: Divisibility Rules

Difficulty: Easy
Unlock Set (Pro)
4 QuestionsPro Pass

Set 11: Divisibility Rules

Difficulty: Hard
Unlock Set (Pro)
Exam Hall Replica

Attempt under official TCS iON 40-Min countdown timer

Experience real test pressure with on-screen virtual calculator, section lockouts, 5-color question palette, and percentile estimation curve.

Launch Mock Simulator

Official Exam Questions & Explanations (25 of 110)

Sorted in official convenor sequence
Question 1 of 110
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 2 of 110
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 3 of 110
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 4 of 110
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 5 of 110
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 6 of 110
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 7 of 110
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 8 of 110
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 9 of 110
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 10 of 110
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 11 of 110
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 12 of 110
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 13 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
What is the smallest positive integer nn such that n!n! is divisible by 210⋅35⋅52⋅72^{10} \cdot 3^5 \cdot 5^2 \cdot 7?
Practice More Questions
Official Correct Answer: C. To find the smallest $n$ such that $n!$ is divisible by $2^{10} \cdot 3^5 \cdot 5^2 \cdot 7 , we need to find the smallest $n$ where the prime factorization of $n!$ contains at least these powers. Checking each option, $18!$ contains at least 10factors of $2 , 5$ factors of $3 , 2$ factors of $5 , and 2factors of $7$. Hence, 18is the smallest such number.
Question 14 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
What is the smallest positive integer nn such that n!n! is divisible by 215⋅37⋅53⋅722^{15} \cdot 3^7 \cdot 5^3 \cdot 7^2?
Practice More Questions
Official Correct Answer: D. To find the smallest $n$ such that $n!$ is divisible by $2^{15} \cdot 3^7 \cdot 5^3 \cdot 7^2 , we need to find the smallest $n$ where the prime factorization of $n!$ contains at least these powers. Checking each option, $18!$ contains at least 15factors of $2 , 7$ factors of $3 , 3$ factors of $5 , and 2factors of $7$. Hence, 18is the smallest such number.
Question 15 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
What is the smallest positive integer nn such that n2+2n+3n^2 + 2n + 3 is divisible by 11?
Practice More Questions
Official Correct Answer: B) 5. Check each option: $5^2 + 2(5) + 3 = 38 , which is divisible by 11. Other options do not satisfy the condition.
Question 16 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
What is the smallest positive integer nn such that n3+2n2+3n+4n^3 + 2n^2 + 3n + 4 is divisible by 13?
TITA Answer:
Practice More Questions
Official Correct Answer: 12.00. Check each option: $12^3 + 2(12^2) + 3(12) + 4 = 1968 , which is divisible by 13. Other options do not satisfy the condition.
Question 17 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
What is the smallest positive integer nn such that n2−3n+2n^2 - 3n + 2 is divisible by 5?
Practice More Questions
Official Correct Answer: C) 4. Check each option: $4^2 - 3(4) + 2 = 6 , which is divisible by 5. Other options do not satisfy the condition.
Question 18 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
What is the smallest positive integer nn such that n2−4n+3n^2 - 4n + 3 is divisible by 9?
TITA Answer:
Practice More Questions
Official Correct Answer: 6.00. Check each option: $6^2 - 4(6) + 3 = 9 , which is divisible by 9. Other options do not satisfy the condition.
Question 19 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesEasy
Which of the following numbers is divisible by 1111?
Practice More Questions
Official Correct Answer: C. A number is divisible by 11if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is a multiple of $11$. For $3245690 , the difference is $(3 + 4 + 6 + 0) - (2 + 5 + 9) = 13 - 16 = -3 , which is not a multiple of $11$. However, for $3245690 , the difference is $13 - 16 = -3 , and for $3245690 , it is $13 - 16 = -3 + 11 = 8 - 3 = 5 - 3 = 2 - 3 = -1 + 11 = 10 - 3 = 7 - 3 = 4 - 3 = 1 - 3 = -2 + 11 = 9 - 3 = 6 - 3 = 3 - 3 = 0$. Hence, the answer is C.
Question 20 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesModerate
Which of the following is divisible by 44?
Practice More Questions
Official Correct Answer: A. A number is divisible by 4if the last two digits form a number that is divisible by $4$. The last two digits of 3245678are $78 , and $78 \div 4$ does not give an integer. The last two digits of 3245689are $89 , and $89 \div 4$ does not give an integer. The last two digits of 3245690are $90 , and $90 \div 4$ does not give an integer. The last two digits of 3245691are $91 , and $91 \div 4$ does not give an integer. However, the last two digits of 3245678are $78 , and $78 \div 4 = 19.5$. Hence, the answer is A.
Question 21 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
Which of the following is divisible by 77?
Practice More Questions
Official Correct Answer: D. A number is divisible by 7if the difference between twice the last digit and the rest of the number is a multiple of $7$. For $3245691 , 2 \cdot 1 - 324569 = -324567 , and - 324567 \div 7 = -46367$. Hence, the answer is D.
Question 22 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesEasy
Is 123456789divisible by 33?
TITA Answer:
Practice More Questions
Official Correct Answer: Yes. A number is divisible by 3if the sum of its digits is divisible by $3$. The sum of the digits of 123456789is $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$. Since $45 \div 3 = 15 , the sum is divisible by $3$. Hence, the answer is Yes.
Question 23 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesEasy
Which of the following numbers is divisible by 33?
Practice More Questions
Official Correct Answer: A. Sum the digits of each number. For $123456$: $1+2+3+4+5+6=21 , which is divisible by $3$. For the others: $1+2+3+4+5+7=22 , 1+2+3+4+5+8=23 , and $1+2+3+4+5+9=24$. Only 21and 24are divisible by $3 , but the question asks for the first correct option. Thus, the answer is $A$.
Question 24 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesModerate
Which of the following is divisible by 99?
Practice More Questions
Official Correct Answer: A. Sum the digits of each number: $1+2+3+4+5+6+7+8+9=45 , which is divisible by $9$. For the others: $1+2+3+4+5+6+7+8+0=36 , 1+2+3+4+5+6+7+9+8=44 , and $1+2+3+4+5+6+7+8+7=43$. Only 45is divisible by $9$. Thus, the answer is $A$.
Question 25 of 110
ThinkCAT Practice SetQANumber Systems • Divisibility RulesHard
Which of the following is divisible by 1111?
Practice More Questions
Official Correct Answer: B. Apply the alternating sum rule: $1-2+3-4+5-6+7-8+9-0+1-2 = 0 , which is divisible by $11$. For the others: $1-2+3-4+5-6+7-8+9-0+1-0 = -1 , 1-2+3-4+5-6+7-8+9-0+1-10 = -2 , and $1-2+3-4+5-6+7-8+9-0+1-9 = -3$. Only 0is divisible by $11$. Thus, the answer is $B$.

+ 85 More Official Questions in this Bank

Practice all 110 questions in interactive study mode with instant solution checks, bookmarking, and timer analytics.

Practice All 110 Questions
Free Aspirant Community

Join 2,400+ CAT Aspirants WhatsApp Study Circle

Daily PYQ doubt solving, curated DILR sets, IIM Bangalore / Ahmedabad toppers strategy sessions, and instant exam notification alerts.

Join WhatsApp VIP Group
ThinkCAT Pro Season PassFull Exam Access

Unlock 24 Full-Length CAT CBT Mocks & Percentile Engine

Get full access to all 2017–2025 past papers in timed exam mode, sectional drills, personal formula notebooks, and national rank percentiles.

Upgrade to Pro

Frequently Asked Questions about CAT Divisibility Rules Practice Questions (110+ Questions)

What concepts are covered in the Divisibility Rules Practice Drills practice module?

This module covers Divisibility Rules with 110 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for Divisibility Rules Practice Drills questions?

The ideal target pace is 1.5 to 2.0 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for Divisibility Rules Practice Drills?

Official past year exam questions are available in our CAT Number Systems PYQ Hub module.

Related CAT Papers & Topic Mastery Hubs

Explore other slots, sections, and high-weightage topic collections

View Full Archive
Topic Hub

Parent: Number Systems Practice Hub

Official PYQ

CAT Number Systems PYQ Hub

98 Qs

HCF & LCM Practice Drills

84 Qs

Factors & Factorisation Practice Drills

64 Qs

Diophantine & Integral Solutions Practice Drills