euclid's algorithm problems ()

Mathematics
Tutor: None Selected Time limit: 1 Day

(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

Studypool's Notebank makes it easy to buy and sell old notes, study guides, reviews, etc.
Click to visit
The Notebank
...
Oct 1st, 2015
...
Oct 1st, 2015
Dec 3rd, 2016
check_circle
Mark as Final Answer
check_circle
Unmark as Final Answer
check_circle
Final Answer

Secure Information

Content will be erased after question is completed.

check_circle
Final Answer