CAT Remainders & modular arithmetic Questions

Remainders describe the residue left after division by a positive integer. Modular arithmetic reduces large expressions to smaller equivalent residues and reveals cycles in powers.

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Key concepts

• Congruence and residue classes
• Cycles of powers
• Remainder constraints

CAT remainders & modular arithmetic practice questions

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Common mistakes

• Dividing congruences without checking whether cancellation is valid.
• Leaving a negative residue when the question asks for the usual non-negative remainder.

Practical solving approaches

• Reduce terms modulo the divisor before multiplying or exponentiating.
• List a power cycle, locate the exponent within it and convert the result to the required remainder range.

Remainders & modular arithmetic FAQs

Can I always cancel a factor in modular arithmetic?

No. Ordinary cancellation modulo m is valid when the factor is coprime to m; otherwise the modulus or possible solutions may change.

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CAT Remainders & modular arithmetic Questions

Remainders describe the residue left after division by a positive integer. Modular arithmetic reduces large expressions to smaller equivalent residues and reveals cycles in powers.

7 free practice questions

Key concepts

• Congruence and residue classes
• Cycles of powers
• Remainder constraints

CAT remainders & modular arithmetic practice questions

Question 1Type in the answerMEDIUM

Question 1

Type in your answer in the input box provided below the question.

Answer the questions independently of each other.

If n is a natural number, then the remainder on dividing (10ⁿ – 7) by 6 is:

Your answer

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Question 2Multiple choiceMEDIUM

Question 2

Select the correct alternative from the given choices.

Answer the questions independently of each other.

What is the remainder when 15 x 13! divides 13! + 20 x 21!?

Choose an answer

A.21B.13!C.12!D.10!

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Question 3Multiple choiceMEDIUM

Question 3

Select the correct alternative from the given choices.

Answer the questions independently of each other.

Let n! = 1 × 2 × 3 × ..... × n for integer n > 1. If p = 1! + (2 × 2!) + (3 × 3!) + ..... + (10 × 10!), then p + 2 when divided by 11! leaves a remainder of

 

Choose an answer

A.10B.0C.7D.1

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Question 4Type in the answerMEDIUM

Question 4

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Answer the questions independently of each other.

A shopkeeper has x number of marbles. If he keeps a multiple of 5 or 9 or 12 or 14 marbles in a box, then he is always left with 3 marbles. What is value of x?

Your answer

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Question 5Multiple choiceHARD

Question 5

Select the correct alternative from the given choices.

Answer the questions independently of each other.

A box contains a number of apples. When the apples are divided into groups of 5, 4 apples are left, when divided into groups of 4, 3 are left, when divided into groups of 3, 2 are left and when divided into groups of 2, 1 is left. What is the minimum possible number of apples?

Choose an answer

A.49B.59C.64D.39

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Question 6Multiple choiceEASY

Question 6

Select the correct alternative from the given choices.

Answer the questions independently of each other.

Choose an answer

A.5B.6C.4D.None of these

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Question 7Multiple choiceHARD

Question 7

Select the correct alternative from the given choices.

Answer the questions independently of each other.

Find the greatest 5 digit number, which on being divided by 5, 6, 7, 8 and 9 leaves 3, 4, 5, 6 and 7 respectively as the remainders.

Choose an answer

A.93240B.98278C.93249D.99261

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Common mistakes

• Dividing congruences without checking whether cancellation is valid.
• Leaving a negative residue when the question asks for the usual non-negative remainder.

Practical solving approaches

• Reduce terms modulo the divisor before multiplying or exponentiating.
• List a power cycle, locate the exponent within it and convert the result to the required remainder range.

Remainders & modular arithmetic FAQs

Can I always cancel a factor in modular arithmetic?

No. Ordinary cancellation modulo m is valid when the factor is coprime to m; otherwise the modulus or possible solutions may change.

More Number Systems practice

• Base systems →
• Factorials & highest power →
• Factors, divisibility, HCF & LCM →
• Number properties & digits →