MAS111_09_assignment_01_hints

# MAS111_09_assignment - NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 1 Hints 1 Prove that the square of any odd

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NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 1 Hints 1. Prove that the square of any odd integer leaves remainder 1 upon di- vision by 8. Hint : Let m be an odd integer, and we can let m = 2 k + 1, where k is an integer. Then m 2 = (2 k + 1) 2 = 4 k ( k + 1) + 1. As k and k + 1 are two consecutive integers, one of them is even, and hence k ( k +1) is also even. Thus 4 k ( k +1) is a multiple of 8. Therefore, when we divide m 2 by 8, the remainder is 1. 2. Prove that the product of two numbers of the form 4 k +3 is of the form 4 k + 1. Hint : As the given two numbers are both of the form 4 k + 3, we can let these two numbers be m = 4 k 1 + 3 and n = 4 k 2 + 3, where k 1 and k 2 are both integers. We need to be aware that we cannot let these two numbers be just 4 k + 3 , because otherwise, these two numbers would be the same. 3. Prove that for natural numbers m,n , m 2 + n 2 is a multiple of 3 if and only if both m and n are multiples of 3. Hint : Consider the remainders when m or n is divided by 3. 4. Prove that there are inﬁnitely many n such that n 2 + 23 is divisible by 24. Hint : Note that n 2 +23 is equal to ( n 2 - 1)+24, so to prove that there are inﬁnitely many n such that n 2 + 23 divisible by 24, we only need to show that there are inﬁnitely many n such that n 2 - 1 divisible by 24. The latter is obviously true, as n 2 - 1 = ( n - 1)( n + 1), and if we

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let n = 24 k + 1, then we will have that n - 1 is always a multiple of 24 (hence n 2 - 1 is a multiple of 24). Note that diﬀerent choices of k will give diﬀerent values of n 2 - 1. 5. (a) Prove that the square of any integer is of the form 4 k or 4 k + 1. (b) Prove that no integer in the sequence
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## This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.

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MAS111_09_assignment - NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 1 Hints 1 Prove that the square of any odd

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