a3 - m 2-n 2 = 88 for which both m and n are positive...

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CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING COMP238 MATHEMATICS FOR COMPUTER SCIENCE I ASSIGNMENT 3 FALL 2011 Due Date: Tuesday November 15 at midnight 1. If A and B are sets and f : A -→ B then for any subset S of B the pre-image of S is f - 1 ( S ) ≡ { a A : f ( a ) S } . Note that f - 1 ( S ) is well de±ned even if f does not have an inverse! Let S and T be subsets of B . Prove that f - 1 ( S T ) = f - 1 ( S ) f - 1 ( T ) , and f - 1 ( S T ) = f - 1 ( S ) f - 1 ( T ) . 2. Show that the function g : ( R - { 1 } ) -→ ( R - { 1 } ) given by g ( x ) = x + 1 x - 1 , is a bijection ( i.e. , one-to-one and onto). 3. i) When an integer b is divided by 12, the remainder is 5. What is the remainder when 8 b is divided by 12? ii) When an integer c is divided by 15, the remainder is 3. What is the remainder when 10 c is divided by 15? 4. Without computing the value of (20!) 2 determine how many zeros are at the end of this number when it is written in decimal form. Justify your answer. 5. Find all solutions to

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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 â‰¡ 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 â‰¥ 2, and a â‰¡ b (mod m ) and c â‰¡ d (mod m ) , then a-c â‰¡ b-d (mod m ) . 9. Prove that if a, b , and m are integers, with m â‰¥ 2, and a â‰¡ b (mod m ), then gcd( a, m ) = gcd( b, m ) . 10. Prove the following by mathematical induction: For all integers n â‰¥ 1, 1 + 6 + 11 + 16 + . . . + (5 n-4) = n (5 n-3) 2 . 11. Prove by induction: For all integers n â‰¥ 1, 5 | (7 n-2 n ) . 12. Prove by induction: For all integers n â‰¥ 2, n ! > 2 n-2 . 2...
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a3 - m 2-n 2 = 88 for which both m and n are positive...

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