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Q47 (CAPF/2018) Science & Technology › Basic Science (Physics, Chemistry, Biology) › Quantitative aptitude topics Answer Verified

Which one of the following is the remainder when 10^20 is divided by 7?

Result
Your answer: —  ·  Correct: B
Explanation

The correct answer is Option 2. To find the remainder when 1020 is divided by 7, we use the concept of modular arithmetic and cyclicity.

First, simplify the base: 10 divided by 7 leaves a remainder of 3. Therefore, 1020 ≡ 320 (mod 7).

Next, we observe the powers of 3 modulo 7 to find a pattern:

  • 31 ÷ 7 = Remainder 3
  • 32 ÷ 7 = Remainder 2
  • 33 ÷ 7 = Remainder 6
  • 34 ÷ 7 = Remainder 4
  • 35 ÷ 7 = Remainder 5
  • 36 ÷ 7 = Remainder 1
The remainders repeat every 6 powers (cyclicity of 6). Alternatively, by Fermat’s Little Theorem, since 7 is prime, 3(7-1) ≡ 36 ≡ 1 (mod 7).

Dividing the exponent 20 by the cycle 6: 20 = (6 × 3) + 2. Thus, 320 ≡ (36)3 × 32 ≡ 13 × 32 ≡ 9 (mod 7). Since 9 divided by 7 leaves a remainder of 2, Option 2 is correct.

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