Unformatted text preview: a (or in b ) are even as well. Therefore, both a and b are perfect squares. 1.72 Let n = p r m , where p is a prime not dividing an integer m ≥ 1. Prove that p³ n p r ´ . Solution. Write a = ³ n p r ´ . By Pascal ’ s formula: a = ³ n p r ´ = n ! ( p r ) ! ( n − p r ) ! . Cancel the factor ( n − p r ) ! and crossmultiply, obtaining: a ( p r ) ! = n ( n − 1 )( n − 2 ) ··· ( n − p r + 1 ). Thus, the factors on the right side, other than n = p r m , have the form n − i = p r m − i , where 1 ≤ i ≤ p r − 1. Similarly, the factors in ( p r ) ! , other than p r itself, have the form p r − i , for i in the same range: 1 ≤ i ≤ p r − 1. If p e  p r m − i , where e ≤ r and i ≥ 1, then p r m − i = bp e ; hence, p e  i ; there is a factorization i = p e j . Therefore, p r − i = p e ( p r − e − j ) ....
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 Fall '11
 KeithCornell
 Prime number

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