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Answers
7) Let us consider
a = n3 – n
a = n (n2 – 1)
a = n (n + 1)(n – 1)
Assumtions:
1. Out of three (n – 1), n, (n + 1) one must be even, so a is divisible by 2.
2. (n – 1) , n, (n + 1) are consecutive integers thus as proved a must be divisible by 3.
From (1) and (2) a must be divisible by 2 × 3 = 6
Thus, n³ – n is divisible by 6 for any positive integer n.
8)Let us consider two odd positive numbers be x and y where
x = 2p + 1 and y = 2q + 1
From question,
x2 + y2 = (2p + 1)2 +(2q + 1)2
= 4p2 + 4p + 1 + 4q2 + 4q + 1
= 4p2 + 4q2 + 4p + 4q + 2
= 4 (p2 + q2 + p + q) + 2
Form above result, we can conclude that x and y are odd positive integer, then x2 + y2 is even but not divisible by four.
Hope that helps you army .
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