Numerical answer
What is the remainder when 7²⁰²¹ is divided by 100?
Show answer and solution
Answer: 7
Look for a small power of 7 that is ≡ 1 (mod 100).
7² = 49, 7⁴ = 2401 ≡ 1 (mod 100).
2021 = 4 × 505 + 1, so 7²⁰²¹ = (7⁴)⁵⁰⁵ × 7 ≡ 1 × 7 = 7 (mod 100).
So the last two digits of 7²⁰²¹ are 07.
Tip: The last two digits of powers of 7 cycle with period 4: 07, 49, 43, 01. Reduce the exponent mod 4.
7² = 49, 7⁴ = 2401 ≡ 1 (mod 100).
2021 = 4 × 505 + 1, so 7²⁰²¹ = (7⁴)⁵⁰⁵ × 7 ≡ 1 × 7 = 7 (mod 100).
So the last two digits of 7²⁰²¹ are 07.
Tip: The last two digits of powers of 7 cycle with period 4: 07, 49, 43, 01. Reduce the exponent mod 4.
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