Cheryl falkowski has a collection of silver spoons from all over the world. she finds that she can arrange her spoons in sets of 7 with 1 left over, sets of 8 with 7 left over, or sets of 15 with 3 left over. if cheryl has fewer than 200 spoons, how many are there?

Since she can arrange it in sets
let N=total number of spoons so
N mod7 1 >7a+1=n where a is a natural num
N mod 8 7>8b+7=n where b is a natural num
N mod 15 3>15c+3=n where c is a natural n
for the first set {8,15,22,29,36,43,50,57,65,71}
for the second set {15,23,31,39,47,55,63,71,79..}
for the third set {18,33,48,63,78,}
and find them manually but what is easier is to solve them using algebra and logic (college level)
N=840m+183 where m is zero or a postive integer
so the only N below 200 is 183
let us check
183/7=26 R 1
183/8=22 R 7
183/15=12 R 3
which means our soultion is correct