math135-a6

math135-a6 - d ( a-a 1 + a 2-a 3 + + (-1) l a l ) . 5. (a)...

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Assignment 6 – Questions 1. Suppose that a,b Z and m,n P . Prove that an bn (mod mn ) if and only if a b (mod m ). 2. Prove that if n 3 (mod 4) then n is not the sum of two squares. 3. Find all possible pairs of digits ( a,b ) such that 99 ± ± 38 a 91 b . 4. Let n = a 0 + a 1 · 1000 + a 2 · 1000 2 + ··· + a l · 1000 l where a l 6 = 0 and for each i we have 0 a i < 1000. Prove that for d = 7, 11 and 13 we have d ± ± n ⇐⇒
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Unformatted text preview: d ( a-a 1 + a 2-a 3 + + (-1) l a l ) . 5. (a) Prove that a positive integer is divisible by 8 if and only if the three-digit integer formed by its nal three digits is divisible by 8. (b) Determine all pairs of digits ( a,b ) such that 27 b 9 a 4 is divisible by 72. 1...
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