CAT 2024 Slot 1 · Quant · Question 10

CAT 2024 Slot 1 previous year question on Remainder Theorem, with the answer and a step-by-step solution.

Remainder TheoremNumber SystemMultiple choiceEasy
When {10}^{100} is divided by 7, the remainder is
  1. A
    3
  2. B
    6
  3. C
    1
  4. D
    4
Pick an option, then check.
Answer & solution

Answer: Option D

Reduce the base first: 10=7+3, so 10\equiv 3 \text{(mod}7).

Hence {10}^{100}\equiv{3}^{100}\text{(mod}7).

List the powers of 3 modulo 7: {3}^{1}\equiv 3, {3}^{2}\equiv 2, {3}^{3}\equiv 6, {3}^{4}\equiv 4, {3}^{5}\equiv 5, {3}^{6}\equiv 1.

The remainders repeat with period 6, since {3}^{6}\equiv 1.

Divide the exponent by the period: 100=6\times 16+4.

So {3}^{100}\equiv{3}^{4}\equiv 4 \text{(mod}7).

The remainder is 4.

Shortcut. By Fermat's little theorem {3}^{6}\equiv 1 \text{(mod}7), so only the exponent modulo 6 matters. Reducing 100 to 4 makes the whole question a single small power.

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