Question on gcd
Find the smallest positive value of so that the .
The answer is 223.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
As gcd(say d) divides 2n+75 and 5n-73, it must divide any linear combination of those two. In particular, it must divide 5×(2n+75)-2×(5n-73) = 521. Clearly, 521 is a prime. So, the choices for d are 1 or 521. As d > 1 is needed, we conclude that d = 521. It implies that 521 divides 2n + 75. As we need a least n, we take 2n + 75 = 521. Solving yeilds n = 223. This n is the least as 2n + 75 is always positive for any positive n and if m < n satisfies the equation i.e 521 divides 2m+75, we would get a contradiction as 2m+75<521.