Interesting Integer N

Substituting $N = 4a + b + 4c$ into $2N = 7a + 6b + 7c$ , we have $8a + 2b + 8c = 7a + 6b + 7c \Rightarrow 4b = a + c$ . Thus $N = 4(a+c) + b = 4(4b) + b = 17b$ and so it must be divisible by $17$ . Since $N \leq 1000$ , thus $1 \leq 17b \leq 1000 \Rightarrow b \leq 58$ . Conversely, given $1 \leq k \leq 58$ , the set $(a, b, c) = (2k, k, 2k)$ will satisfy the given conditions. Hence there are $58-1 + 1 = 58$ integer values of $N$ that satisfy the given conditions.