Numerical answer
Find the remainder when 7⁴⁵ is divided by 11.
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Answer: 10
By Fermat's Little Theorem, since 11 is prime and gcd(7,11)=1, 7¹⁰ ≡ 1 (mod 11). Write 45 = 4×10 + 5, so 7⁴⁵ ≡ 7⁵ (mod 11). Compute powers of 7 mod 11: 7¹=7, 7²=49≡5, 7³≡35≡2, 7⁴≡14≡3, 7⁵≡21≡10. So the remainder is 10.
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