SOLUTION: Dear sir/madam, I am stucked with the following problems and hope that you could guide me. 1. (a) Given c>0, prove that lcm(ac,bc)= clcm(a,b) (b) Prove that one of any m

Algebra ->  Algebra  -> Divisibility and Prime Numbers -> SOLUTION: Dear sir/madam, I am stucked with the following problems and hope that you could guide me. 1. (a) Given c>0, prove that lcm(ac,bc)= clcm(a,b) (b) Prove that one of any m      Log On

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Question 136217This question is from textbook
: Dear sir/madam,
I am stucked with the following problems and hope that you could guide me.
1. (a) Given c>0, prove that lcm(ac,bc)= clcm(a,b)
(b) Prove that one of any m consecutive integers must be divisible by m.
Thank you very much.
Best wishes,
Mr Tan GuodongThis question is from textbook

Answer by solver91311(12126) About Me  (Show Source):
You can put this solution on YOUR website!
I'm not at all sure how to even approach the first one. Perhaps someone else can help you.

But here is a rather intuitive discussion of the second problem:

If some integer p is divided by another integer m, p%3E=m, using integer division, then there are exactly m-1 possible non-zero remainders. If given m consectutive integers, dividing each by m will result in m different remainders. Since there are only m-1 non-zero remainders available, one of the m remainders must be 0, hence one of the m integers is evenly divisible by m.