Problem: The number of solution-pairs in positive integers of the equation 3x+5y=501 is:
Answer Choices:
A. 33
B. 34
C. 35
D. 100
E. none of these.
Solution:
3x=501−5y. For x to be a positive integer 3501−5y>0∴5y<501 and y≤100. Also x=167−y−32y; for integral x,y must be a multiple of 3 , that is, y=3k. Since y≤100, k=1,2,…,33.
or
In Number Theory it is shown that if x0,y0 or is one solution of 3x+5y=501, then other solutions are x=x0−d5t,y=y0+d3t where t is an integer and d is the greatest common divisor of 3 and 5 , so that, in this case, d=1. An obvious solution of the given equation is x=167,y=0. Therefore, other solutions are x=167−5t,y=0+3t. Since x=167−5t>0,t<5167 so that t=1,2,…,33.