Expert: Sherry Wallin - 12/21/2009

Question
Hello,please help prove this exercises fully please explain

1.For any positive integer n,find lcm(n,n+1)

2.True or false (if false give a counterexample)
If b is the largest square divisor of c and a2|c, then a|b ?


Thank you

Answer
Anndy~
    
#1 Regardless what n is, n and n=1 are right next to each other and one is odd and the other is even. They will have no factors in common so their least common multiple is n(n+1). I'm not sure what method you use to find the lcm but one way is to prime factor and then use all the prime factors the most times they occur in either. Since the two numbers have no factors in common all of each of their prime factors will be used to find their lcm.

#2 True. Let b be the largest divisor of c, so b|c -> c = bk for some integer k. And a^2|c -> c = a^2m for some integer m. Note that both k and m divide c too. You have two representations for c, set them equal to each other: a^2m=bk. Notice that bk is c and we know a^2 divides c = bk. If a^2 divides any number certainly 'a' will divide that number thus a|b.

Math Prof