For what least value of n, 2 2n +2n is completely divided by 6, where n is an integer?

Correct — C. Factor the expression before testing anything: 2²ⁿ + 2ⁿ = 2ⁿ(2ⁿ + 1).

n = 0 → 1 × (1 + 1) = 2, and 2 is not a multiple of 6.
n = 1 → 2 × (2 + 1) = 6, which 6 divides exactly.

No smaller integer can work: for negative n the expression is a fraction (n = −1 gives 3⁄4), and n = 0 has just failed.

So the least value is n = 1 — option (c).

Why the others are wrong

• (a)
  3 — n = 3 does work: 2³ × (2³ + 1) = 8 × 9 = 72, a multiple of 6. But the stem asks for the least such n, and 1 is smaller.
• (b)
  2 — n = 2 gives 2² × (2² + 1) = 4 × 5 = 20. It is even, but 20 carries no factor of 3, so 6 does not divide it.
• (d)
  0 — n = 0 gives 1 × 2 = 2. Divisible by 2 and by nothing else that matters here — the factor of 3 is missing.

6 = 2 × 3, so the test is really two tests: divisible by 2 and divisible by 3.

Writing 2²ⁿ + 2ⁿ as 2ⁿ(2ⁿ + 1) separates them. The 2ⁿ supplies the factor of 2 once n ≥ 1, and since 3 never divides a power of 2, the 3 has to come out of 2ⁿ + 1.

Powers of 2 leave remainders 2, 1, 2, 1 … on division by 3 as n runs 1, 2, 3, 4 …, so 2ⁿ + 1 is a multiple of 3 exactly when n is odd.

Key facts

• 6 divides a number exactly when 2 and 3 both divide it.
• 2²ⁿ + 2ⁿ factorises as 2ⁿ(2ⁿ + 1).
• 3 never divides a power of 2, so the factor of 3 must come from 2ⁿ + 1.
• 2ⁿ + 1 is a multiple of 3 exactly for odd n, giving 6 at n = 1 and 72 at n = 3.

Common traps

• Substituting from the largest option downwards and stopping at the first success.
• Checking evenness only — 20 at n = 2 is even and still fails.
• Reading the printed '2 2n' as 22n or as 2 × 2n.

No directly related past PYQ was found.