Question

State all the possible values for gcd(m,m+6) where m is an integer.

Answer 1

Let

d=gcd(m,m+6)

d/m and d/(m+6).

d/((m+6)-m)

d/6 ........(1)

So the positive numbers dividing 6 are 1,2,3,6

(When we talk about GCD,then it is obvious that we are talking about positive integers.)

Now taking m=1 then m+6=7

Then gcd(1,7) =1

So 1 is the possible case for gcd(m,m+6)

Also taking m=2 then ,m+6=8

Then gcd(2,8)=2.

Hence 2 is also possible case for gcd(m,m+6).

Now take m=3,then m+6=9,then

gcd(3,9)=3

So 3 is also possible case for gcd(m,m+6).

Taking m=6,then m+6=12

Then gcd(6,12)=6.

Hence 6 is also possible case for gcd(m,m+6).

Also from equation(1),gcd(m,m+6) can't be greater than 6.

Hence possible values for gcd(m,m+6) are 1,2,3 & 6.

Hence the solution.

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