There are a number of ways to approach this problem, including writing all the numbers that match one of the arrangements, such as for sets of 15:
.. 14, 29, 44, 59, 74, 89, 104, 119, 134, 149, 164, 179, 194 (separated by 15)
Modulo 8, these numbers are
.. 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2
This reduces the numbers of interest to those that are 3 modulo 8:
.. 59, 179 (separated by 8*15 = 120)
Looking at these modulo 7, we have
.. 3, 4
pinpointing 179 as the number of spoons in Cheryl's collection.
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The attachment shows a solution via graphing calculator. We simply added the absolute values of the differences of the remainders from the desired values and looked for where that sum was zero. This method, too, pointed to 179 as the solution to the problem.
.. 14, 29, 44, 59, 74, 89, 104, 119, 134, 149, 164, 179, 194 (separated by 15)
Modulo 8, these numbers are
.. 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2
This reduces the numbers of interest to those that are 3 modulo 8:
.. 59, 179 (separated by 8*15 = 120)
Looking at these modulo 7, we have
.. 3, 4
pinpointing 179 as the number of spoons in Cheryl's collection.
_____
The attachment shows a solution via graphing calculator. We simply added the absolute values of the differences of the remainders from the desired values and looked for where that sum was zero. This method, too, pointed to 179 as the solution to the problem.