Remainder Theorem Questions with Solutions | Set 1 (Q1 to Q10)

This set contains Remainder Theorem Questions with Solutions — Set 1 (Q1 to Q10) covering a mix of question types and difficulty levels — from basic to advanced — exactly as asked in real competitive exams.

Solutions are written in a simple, step-by-step notebook style for easy self-study and quick understanding. Each solution is broken down step by step so even the toughest question feels easy. These questions are hand-picked for students preparing for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and all campus placement aptitude tests. International students preparing for GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests will find these equally useful.

✏️ Attempt each question on your own first — then check the solution below.

1. Find the remainder of \(\frac{{18 + 17 – 15 + 130 + 138 – 178}}{{16}}\)

2. Find the remainder of \(\frac{{1576 + 3039 + 1214}}{{15}}\)

3. Find the remainder of \(\frac{{71 + 80 + 89 – 107}}{9}\)

4. Find the remainder of \(\frac{{341 \times 342 \times 343 \times 344}}{{13}}\)

5. Find the remainder of \(\frac{{{{243}^{243}}}}{{244}}\)

6. Find the remainder when 123456……..61 digits is divided by 16.

7. Find the remainder when 12345……….. till 31 digits is divided by 8.

8. Find the remainder of \(\frac{{{8^{77}}}}{{17}}\)

9. Find the remainder when 1! + 2! + 3! + ……….. + 100! is divided by 
(i) 5          (ii) 6        (iii) 12

10. Find the remainder when 84 × 86 × 399 × 432 is divided by 380.

1. Find the remainder of \(\frac{{18 + 17 – 15 + 130 + 138 – 178}}{{16}}\)
Sol:

3. Find the remainder of \(\frac{{71 + 80 + 89 – 107}}{9}\)
Sol:

4. Find the remainder of \(\frac{{341 \times 342 \times 343 \times 344}}{{13}}\)
Sol:

5. Find the remainder of \(\frac{{{{243}^{243}}}}{{244}}\)
Sol:

6. Find the remainder when 123456……..61 digits is divided by 16.
Sol:
To get remainder of division by 16 we need last 4 digits.
⟹ 1 ⎯ 9 ⟶ 9 digits
remaining digits = 61 – 9 = 52
starting from 10 further numbers have 2 digits
∴ \(\frac{{52}}{2}\) = 26 numbers are required of which first number is 10
∴ last number = 10 + (26 – 1).1 = 35
∴ last 4 digits are 3435
Remainder = \(\frac{{3435}}{{16}}\) = 11    Answer

7. Find the remainder when 12345……… till 31 digits is divided by 8.
Sol:
To know the remainder of division by 8 we only need last 3 digits.
⟶ 1 to 9 ⟹ 9 digits
Remaining digits = 31 – 9 = 22
each number after 9 will have 2 digits
∴ number required = \(\frac{{22}}{2}\) = 11
∴ last number = 10 + (11 – 1).1 = 20
∴ last 3 digits will be 920
Now check Remainder ⟹ 
\(\frac{{920}}{8}\) = 0     Answer

8. Find the remainder of \(\frac{{{8^{77}}}}{{17}}\)
Sol:
Method(1):
\(\frac{{{8^{77}}}}{{17}} = \frac{{{2^{231}}}}{{17}} = \frac{{{{\left( {{2^4}} \right)}^{57}}{{.2}^3}}}{{17}}\)

Method(2):
\(\frac{{{8^{77}}}}{{17}}\)
Φ(17) = 16 & 8 and 17 are co-prime
∴ \(\frac{{{8^{16}}}}{{17}}\) = 1 ⟵ Remainder

9. Find the remainder when 1! + 2! + 3! + ……….. + 100! is divided by
(i) 5          (ii) 6        (iii) 12
Sol:

10. Find the remainder when 84 × 86 × 399 × 432 is divided by 380.
Sol:

✅ Well done on completing Set 1!

Continue practising with Remainder Theorem Questions 11 to 20 → Set 2 or revisit the Remainder Theorem Concept Page to strengthen your formulas and tricks before moving ahead.

Consistent practice is the key to mastering remainder theorem for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and international exams including GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests. Want to understand the concept better? Read about Remainder Theorem on Wikipedia before attempting the next set.

This page is part of our complete series of remainder theorem questions with solutions for competitive exams — covering every question type from basic to advanced so you can build speed, accuracy and confidence. Practising these questions regularly will also strengthen your core remainder theorem concept before your exam day.