Can all other numbers be expressed?

Short Solution:

A quick glance at the options shows that for $8$ , $y$ cannot be a non-negative integer (here, it is $\frac{1}{3}$ ).

Complete Solution:

The number in question, when divided by $7$ , must leave a whole number with a remainder divisible by $3$ . We can convert the options like so:

$15 \equiv 1 \pmod 7$

$8 \equiv 1 \pmod 7$

$11 \equiv 4 \pmod 7$

$13 \equiv 6 \pmod 7$

Modular Arithmetic

From first glance, $15, 8, 11$ do not satisfy the requirements. ( $13$ is possible since $6$ is divisible by $3$ .) However, we also need to take all of these $\pmod 3$ . Here we go:

$15 \equiv 0 \pmod 3$

$8 \equiv 2 \pmod 3$

$11 \equiv 2 \pmod 3$

As can be seen, $15$ is another possible choice (it can be made using $7 \times 0+3 \times 5$ ). We are left with $11$ and $8$ . The smallest of them is $\boxed 8$ .

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Alternatively, use the Postage Stamp Problem / Chicken McNugget Theorem . By direct application of it, we find that every number greater than $12$ can be expressed in the form $7x+3y$ . Of the options, we only have $8$ and $11$ left. We can then easily find that $8$ is the correct answer.

given that x and y are positive, 8 is the least number which cannot be expressed as 7x+3y.