tut12-large - also numbers u,v such that gcd (168 , 456) =...

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Problems to Week 13 Tutorial — MACM 101 1. Determine the quotient q and the remainder r for each of the following, where a is the dividend and b is the divisor. (a) a = 123, b = 17; (b) a = - 115, b = 12; (c) a = 0, b = 42; (d) a = 434, b = 31. 2. Write each of the following 10-base numbers in base 2 and 16 (a) 137; (b) 6243; (c) 12345. 3. Convert each of the following binary numbers to base 10 (a) 11001110; (b) 00110001. 4. For positive integers a,b and d = gcd( a,b ), prove that gcd ± a d , b d ² = 1 . 5. Let n be a positive integer. Prove that gcd( n,n + 2) equals 1 or 2. 6. Find the greatest common divisor of 168 and 456, and
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Unformatted text preview: also numbers u,v such that gcd (168 , 456) = 168 u + 456 v . 7. Find the greatest common divisor and the least com-mon multiple of 630 and 40452. 8. Find the prime factorization of (a) 148500; (b) 7114800; (c) 7882875. 1 9. How many positive divisors are there for n = 2 14 3 9 5 8 7 10 11 3 13 5 37 10 . 10. A perfect square is a number n such that n = k 2 for some integer k . Determine the smallest perfect square that is divisible by 7!. 2...
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tut12-large - also numbers u,v such that gcd (168 , 456) =...

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