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**Prove that of 25 is subtrated from the square of an odd integer greater than 5, the resulting number is always divisible by 8.**

Solution:

Let S (n) = n2 – 25

For n = 1, S (1) = (1 )2 – 25 = -24 which is clearly divisible by 8.

Thus the first condition is satisfied as S(1) is true.

Let us assume that S(n) is true for n = 2k+1 Belonging to Odd integers greater than 5, that is,

S(2k+1) = (2k+1)2 - 25 is divisble by 8

= 4k2 + 4k + 1 -25

= 4k2 + 4k - 24

= 4(k2 + k - 6)

= 4(k-2)(k+3) ------ (A)

We can see that (A) is clearly divisible by 8, condition being that k is an odd integer which is bigger than 5. Since both conditions are satisfied, hence by mathematical induction we have proven that S(n) is divisble by 8 for all integers bigger than 5.