If n is a natural number, then what is the number of distinct remainders of when divided by 4?

Correct Option

1 (only remainder 0)

Explanation

Short explanation:

• Write the numerator as n^{88}(n^{12}-n^8+n^4-1).
• Consider n modulo 4:
  • n ≡ 0 ⇒ n^{88} ≡ 0 (mod 4) ⇒ whole expression ≡ 0 (mod 4).
  • n ≡ 1 ⇒ each power ≡ 1 so bracket = 1−1+1−1 = 0 ⇒ whole expression ≡ 0 (mod 4).
  • n ≡ 2 ⇒ n^{88} ≡ 0 (mod 4) (since 2^2 ≡ 0) ⇒ whole expression ≡ 0 (mod 4).
  • n ≡ 3 ⇒ 3 ≡ −1 so each even power ≡ 1, bracket = 1−1+1−1 = 0 ⇒ whole expression ≡ 0 (mod 4).
• Hence for every natural number n the expression is divisible by 4; the only remainder is 0. Therefore the number of distinct remainders is 1.