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# sols3 - MATH 135 Solutions to Assignment 3 W10 This...

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MATH 135 Solutions to Assignment 3 W10 This assignment is due at 8:30am on Wednesday January 27, in the drop box outside the Tutorial Centre, MC 4067 Remember to show all of your computational work. The following list of the first few primes may be useful: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, . . . 1. [ 8 marks, distributed as (a) 4, (b) 2, (c) 2 ] (a) Find the prime factorization of a = 10! and b = 2 20 - 1. Hint : For b , recall that ( x 2 - y 2 ) = ( x + y )( x - y ). Answer : a = 10! = 2 8 · 3 4 · 5 2 · 7, and b = (2 5 - 1)(2 5 + 1)(2 10 + 1) = 31 · 33 · 1025 = 31 · (3 · 11)(5 2 · 41) = 3 · 5 2 · 11 · 31 · 41 (b) Find the number of positive factors of a and b . Hint : Given the prime factorization of x , x = p d 1 1 p d 2 2 p d 3 3 . . . p d n n , the number of positive factors of x is given by ( d 1 + 1)( d 2 + 1)( d 3 + 1) . . . ( d n + 1). Answer : We use the formula: thus a has (8+1)(4+1)(2+1)(1+1) = 270 factors, and b has (1 + 1)(2 + 1)(1 + 1) 3 = 48 factors (c) Find the greatest common divisor and the least common multiple of a and b . Answer : By part (a), gcd( a, b ) = 3 · 5 2 = 75, and lcm ( a, b ) = 2 8 · 3 4 · 5 2 · 7 · 11 · 31 · 41 2. [

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sols3 - MATH 135 Solutions to Assignment 3 W10 This...

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