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MAS111_09_assignment_02_hints

# MAS111_09_assignment_02_hints - NANYANG TECHNOLOGICAL...

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Unformatted text preview: NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 2 HINTS 1. For any positive odd numbers m,n , prove that m 2 + n 2 is even, but not a multiple of 4. Hint : Let m = 2 k + 1 and n = 2 l + 1, and calculate m 2 + n 2 . 2. Prove that for any natural number n, 3 does not divide n 2 + 1. Hint : Consider the cases when n = 3 k + i for integer k and i = 0 , 1 , 2. 3. Prove that √ 5 + √ 7 is an irrational number. Hint : Prove it by contradiction. Suppose that √ 5+ √ 7 is a rational number then √ 5- √ 7 is also rational (why?). Now √ 5 = 1 2 (( √ 5 + √ 7) + ( √ 5- √ 7)) is again rational, which is not true (you may prove here that √ 5 cannot be rational by one more contradiction proof). 4. Prove that for any natural number n , 2 2 n- 1 is divisible by 3. Hint : Prove it by induction. Note that 2 2( k +1)- 1 = 2 2( k +1)- 2 2 k + 2 2 k- 1 = 3 · 2 2 k + 2 2 k- 1....
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MAS111_09_assignment_02_hints - NANYANG TECHNOLOGICAL...

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