Find the remainder when $2 3^{32}$ divided by 529.

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UP Police SI (दरोगा) Official PYP (Held On: 12 Nov 2021 Shift 3)

Answer 1

Given:

We need to find the remainder when 23³² is divided by 529.

Concept Used:

Euler's Totient Theorem: If a and n are coprime, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function.

For n = p^k where p is a prime number, φ(n) = p^k - p^k-1.

Calculation:

Calculate φ(529)

Since 529 = 23², we have φ(529) = 23² - 23¹ = 529 - 23 = 506.

Since 23 and 529 are not coprime, we cannot directly apply Euler's theorem. However, we can rewrite 23³² as (23²)¹⁶ = 529¹⁶.

We want to find the remainder of 529¹⁶ when divided by 529.

Since 529¹⁶ is a multiple of 529, the remainder is 0.

∴ The remainder when 23³² is divided by 529 is 0.