A) Find the L.C.M of 6 and 9. b) Find the smallest number when divided by either 7 or 8 the remainder is 4.

$$6 = 2 \times 3$$ $$9 = 3 \times 3 = 3^2$$

$$\text{L.C.M}(6, 9) = 2^1 \times 3^2 = 2 \times 9 = 18$$

The L.C.M of 6 and 9 is $\boxed{18}$.

Since 7 and 8 are coprime (they share no common factors other than 1), their L.C.M is their product.

$$\text{L.C.M}(7, 8) = 7 \times 8 = 56$$

This means the number is 4 more than a multiple of the L.C.M of 7 and 8.

$$\text{Smallest number} = \text{L.C.M}(7, 8) + 4$$ $$= 56 + 4$$ $$= 60$$

The smallest number is $\boxed{60}$.