Question

The least common multiple of nonzero integers a and b is the smallest positive integer m...

The least common multiple of nonzero integers a and b is the smallest positive integer m such that a | m and b | m; m is usually denoted [a,b]. Prove that

[a,b] = ab/(a,b) if a > 0 and b > 0.

Math   Algebra   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

Proof:

If  , we can write   and .

divides    . Similarly , divides .  

Hence   is a common multiple of   and   and  .

Let   be any positive common multiple  of   and   so that   and   for some  

Since    , there are integers   such that  .

Hence ,   divides

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let a, b be integers with not both 0. Prove that hcf(a, b) is the smallest...
Let a, b be integers with not both 0. Prove that hcf(a, b) is the smallest positive integer m of the form ra + sb where r and s are integers. Hint: Prove hcf(a, b) | m and then use the minimality condition to prove that m | hcf(a, b).
Let m and n be positive integers and let k be the least common multiple of...
Let m and n be positive integers and let k be the least common multiple of m and n. Show that mZ intersect nZ is equal to kZ. provide justifications pleasw, thank you.
Let m and n be positive integers and let k be the least common multiple of...
Let m and n be positive integers and let k be the least common multiple of m and n. Show that mZ intersect nZ is equal to kZ. provide justifications please, thank you.
Prove that any two nonzero elements of a UFD have a least common multiple, and describe...
Prove that any two nonzero elements of a UFD have a least common multiple, and describe the least common multiple in terms of the prime factorizations of the two elements.
9. Let a, b, q be positive integers, and r be an integer with 0 ≤...
9. Let a, b, q be positive integers, and r be an integer with 0 ≤ r < b. (a) Explain why gcd(a, b) = gcd(b, a). (b) Prove that gcd(a, 0) = a. (c) Prove that if a = bq + r, then gcd(a, b) = gcd(b, r).
(a) If a and b are positive integers, then show that lcm(a, b) ≤ ab. (b)...
(a) If a and b are positive integers, then show that lcm(a, b) ≤ ab. (b) If a and b are positive integers, then show that lcm(a, b) is a multiple of gcd(a, b).
Let m be a composite positive integer and suppose that m = 4k + 3 for...
Let m be a composite positive integer and suppose that m = 4k + 3 for some integer k. If m = ab for some integers a and b, then a = 4l + 3 for some integer l or b = 4l + 3 for some integer l. 1. Write the set up for a proof by contradiction. 2. Write out a careful proof of the assertion by the method of contradiction.
Prove that for positive integers a and b, gcd(a,b)lcm(a,b) = ab. There are nice proofs that...
Prove that for positive integers a and b, gcd(a,b)lcm(a,b) = ab. There are nice proofs that do not use the prime factorizations of a and b.
prove: H contains of positive integer H=tz for t=least positive integer in H.
prove: H contains of positive integer H=tz for t=least positive integer in H.
Three positive integers a, b and c have, in pairs, highest common factors (a,b) = 24,...
Three positive integers a, b and c have, in pairs, highest common factors (a,b) = 24, (b,c) = 198 and (a,c) = 210. (a) What is the highest common factor of a, b and c? (b) Find the smallest values of a, b and c which satisfy the given criteria.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT