Question 4726: Show that 5n+3 and 7n+ 4 are relatively prime for all n
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! Use the fact about gcd:
d=(a,b) if and only if there exist integer p, q such that
pa + qb = d.
Hence, (a,b) = 1 if and only if there exist integer p, q such that
pa + qb = 1.
Now, 3(5n+3)-2(7n+ 4) = 15n + 9 -14n- 8 = 1
So, (5n+3, 7n+ 4) = 1 ie (5n+3 ) and (7n+ 4) are relative prime for all n.
Another way of proof:
if d =(5n+3, 7n+4), consider 7(5n+3)-5(7n+4) = 1,
since d is a divisor of 7(5n+3)-5(7n+4),
d must be 1 and so (5n+3 ) and (7n+ 4)are relative prime.
Kenny
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