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Unformatted text preview: m 2-n 2 = 88, for which both m and n are positive integers. 6. Prove that if n is an odd positive integer then n 2 1 (mod 8) . 1 7. Prove that for all integers n we have n 3 = 9 k , or n 3 = 9 k + 1 , or n 3 = 9 k-1 , for some integer k . 8. Prove that if a, b, c, d , and m are integers, with m 2, and a b (mod m ) and c d (mod m ) , then a-c b-d (mod m ) . 9. Prove that if a, b , and m are integers, with m 2, and a b (mod m ), then gcd( a, m ) = gcd( b, m ) . 10. Prove the following by mathematical induction: For all integers n 1, 1 + 6 + 11 + 16 + . . . + (5 n-4) = n (5 n-3) 2 . 11. Prove by induction: For all integers n 1, 5 | (7 n-2 n ) . 12. Prove by induction: For all integers n 2, n ! > 2 n-2 . 2...
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- Spring '11
- Software engineering