Find the smallest positive integer k for which 360k is a multiple of 2750?

Answer 1

k=275

First find the prime factorization of 2750 and 360:

2750=2*53*11

360=23*3*5

Now you want the smallest positive integers, k and m, such that:

23*3*5*k=2*53*11*m

In order for the two sides of the equation to be equal, the right-hand side of the equation must contain the prime factors on the left-hand side to the same power as the prime factors on the left-hand side. The smallest integer value of m for which this is true is:

m=22*3

Now solve for k:

23*3*5*k=2*53*11*22*3

k=(2*53*11*22*3)/(23*3*5)=52*11=275