Question

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explain and give the answers for these two questions...

1. use euclid's division lemma to show that any positive ood integer is of the form 6q+1 or 6q+3 or 6q+5, where q is some integer.

2. use euclid's division lemma to to show that cube of any positive integer is of the form 9m, 9m+1, or 9m+8.

Answer 1

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Answer 2

1. let a be an odd positive integer

Let us now apply division alogarithm with a and b = 6

Therefore 0 <r <6, the possible remainders are 0,1,2,3,4and 5.

I.e.,a can be 6q or 6q +1 or 6q+2 or 6q+3 or 6q+4 or 6q+5, where q is the quotient

But a is taken as an odd number.

Therefore, a can't be 6q ,6q+2,6q+4

Hence any odd integer is in the form 6q+1 , 6q+3, 6q+5

2. let a be the cube of a positive integer.

Applying Euclid's division lemma for a and b =9

a=bm+r where m is quotient and r is remainder , where 0 <r <9.

Therefore a can't be of the form 9m,9m+1, 9m+2............or 9m+8

Hence cube of any positive integer is of the form 9m,9m+1.....or 9m+8