euclid's algorithm problems ()

label Mathematics
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(a) prove that if gcd(a,b)=k and gdc(b,c)=k, then gcd(a,c)>=k.

(b) determine a formula for the least common multiple of two integers whose difference is a prime number.(hint: consider two cases)

Oct 1st, 2015

Thank you for the opportunity to help you with your question!

(a) Let a = k*a1, b = k*b1, c = k*c1, where gdc(a1, b1) = gdc(b1, c1) = 1

Therefore, gdc(a1, c1) >= 1, which means

gcd(a,c) = gdc(k*a1, k*c1) = k*gdc(a1, c1) >= k*1 = k

(b) Let the two numbers be a, a + p, where p is prime.

Case 1: if a is divisible by p then gcd(a, a+p) = p. Therefore, lcm(a, a+p) = a*(a+p)/gcd(a, a+p) = a(a+p)/p

Case 1: if a is NOT divisible by p then gcd(a, a+p) = 1. Therefore, lcm(a, a+p) = a*(a+p)/gcd(a, a+p) = a(a+p)/1 = a(a+p)

** P/S: Please remember to give me a review later on once the answer is correct. I would be highly appreciate it. Thank you. **
Oct 1st, 2015

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