If m is even, then (8 m – 1) is divisible by:

Correct — D. Read the term as 8ᵐ − 1. The exponent is flattened to 8 m in the response-sheet capture, and the plain product 8m − 1 gives 15, 31 and 47 at m = 2, 4, 6, which no option divides.

m is even, so write m = 2k
8ᵐ = 8²ᵏ = 64ᵏ
aⁿ − bⁿ is always divisible by a − b, so 64ᵏ − 1 is divisible by 64 − 1 = 63

Check it: 8² − 1 = 63, and 8⁴ − 1 = 4095 = 63 × 65 → option (d)

Why the others are wrong

• (a)
  8 — 8 — 8ᵐ is itself a multiple of 8, so 8ᵐ − 1 is one less than a multiple of 8 and can never be divisible by it, for odd m or even.
• (b)
  65 — 65 — 65 divides 64ᵏ − 1 only when k is even, that is when m is a multiple of 4. At m = 2 the value is 63, and 65 does not divide 63.
• (c)
  42 — 42 — 42 is even, but 8ᵐ − 1 is always odd, one less than an even number. An odd number has no even divisor at all.

Two factorisation facts settle almost every question of the form aⁿ ± 1 is divisible by what.

The first: aⁿ − bⁿ is divisible by a − b for every positive integer n. Put b = 1 and it reads aⁿ − 1 is divisible by a − 1.

The work is in choosing a. Written as 8ᵐ, the base gives 8 − 1 = 7, so 7 divides it for every m. Regrouping an even exponent as 64ᵏ gives the stronger 64 − 1 = 63, and that is exactly what the word even buys you.

Key facts

• aⁿ − bⁿ is divisible by a − b for every positive integer n.
• 8ᵐ − 1 is divisible by 7 for every m, because 8 − 1 = 7.
• When m is even, 8ᵐ is a power of 64, so 8ᵐ − 1 is divisible by 64 − 1 = 63.
• 8² − 1 = 63 and 8⁴ − 1 = 4095 = 63 × 65.

Common traps

• Reading 8 m as the product 8 × m, which leaves the question with no correct option
• Stopping at 7, the divisor that works for every m, and missing what even adds
• Choosing 65, which needs m to be a multiple of 4 rather than merely even

No directly related past PYQ was found.