Question

Prove by method of induction, for all n ∈ N
$(2^{4n}-1)$ is divisible by 15.

Answer 1

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Answer 2

Answer:

Hence 2⁴ⁿ - 1   is divisible by 15

Step-by-step explanation:

p(n) = 2⁴ⁿ - 1   is divisible by 15

p(1) = 2⁴ - 1  = 16 - 1 = 15    is divisible by 15

p(2) = 2⁴ˣ² - 1 = 2⁸ - 1 = 256 - 1 = 255  = 15 * 17    is divisible by 15

Let say

p(a) = 2⁴ᵃ - 1   is divisible by 15

then

2⁴ᵃ - 1  = 15 k

=> 2⁴ᵃ = 15k + 1

p (a + 1) = 2⁴⁽ᵃ⁺¹⁾ - 1

= 2⁴ᵃ ⁺ ⁴ - 1

= 2⁴ᵃ * 2⁴ - 1

= 16 (15k + 1) - 1

= 16 * 15k + 16 - 1

= 16* 15k + 15

= 15 ( 16k + 1)

Divisible by 15

Hence 2⁴ⁿ - 1   is divisible by 15