Question

what is the minimum value of abs(578m-910n-541)​

Answer 1

we have to find minimum value of abs(578m - 910n - 541) or, min |578m - 910n - 541| where m and n are integers.

prime factors of 578 = 2 × 17 × 17

prime factors of 910 = 2 × 5 × 7 × 13

so, gcd{578, 910} = 2

so, we can write it, 578m - 910n = 2(278m - 455n)

now, |578m - 910n - 541| converts into |2(278m - 455n) - 541|

Let us consider (278m - 455n ) = x

then, |2(278m - 910n) - 541| = |2x - 541|

if m and n are integers then, (278m - 455n) is also a integers value .

but we have to find minimum value so, we have to put value of x in such a way that |2x - 541| will be minimum.

if we put x = 271

then, 2 × 271 - 541 = 1( minimum absolute value.

here we also see , if we take x = 278m - 455n = 271

then, m = 34 (integer) and n = 21 ( integer).

hence, value of abs(578m - 910n - 541) = 1