5 Let m n and d be integers Show that a n n b d m m n d n c d n n d d n dd n d

5 let m n and d be integers show that a n n b d m m n

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(For all questions show your working.) 5. Let m , n and d be integers. Show that (a) n | n . (b) d | m, m | n d | n . (c) d | n, n | d d = ± n . (d) d | n, d | m d | ( xn + ym ) , for all integers x and y . 6.Ifnandd6= 0are integers, show that there exist integersqandrsuch thatn=qd+rand0r≤ |d|. 7. In each case, compute gcd( m, n ) and express it as a linear combination of m and n . (a)m= 72,n= 42(b)m= 327,n= 54(c)m= 377,n= 29(d)m= 72,n=-175 8. If gcd( m, n ) = 1 and gcd( k, n ) = 1 , show that gcd( mk, n ) = 1 . 9. If m | m 1 and n | n 1 , show that gcd( m, n ) | gcd( m 1 , n 1 ) . 10. Let d = gcd( m, n ) . If k | d , k 1 , show that gcd m k , n k = d k 11.Find thegcdand thelcmof the following numbers: 3
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I.3. Integers modulo n (For all questions show your working.) 12.Prove Wilson’s Theorem:(p-1)!≡ -1(modp) 13. In each case determine whether the statement is true or false.
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  • Fall '15
  • 3k, 2k, 5k, I.2. Divisibility

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