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$x$ | $11x$ | Remainder when divided by $7$ |
---|---|---|
0 | 0 | 0 |
1 | 11 | 4 |
2 | 22 | 1 |
3 | 33 | 5 |
4 | 44 | 2 |
5 | 55 | 6 |
6 | 66 | 3 |
7 | 77 | 0 |
8 | 88 | 4 |
9 | 99 | 1 |
This means the only possible value of $x$ is $4$.
Substituting this in, $44 + 7y = 100$. Subtracting 44 gives $7y = 56$, and dividing by $7$ gives $y = 8$.
Therefore Tom bought $4$ stamps and Sophie bought $8$ stamps.