In the diagram, the number line is marked at consecutive integers, but the numbers themselves are not shown. The four larger dots represent two numbers that are multiples of 3 and two numbers that are multiples of 5 . Which point represents a number which is a multiple of 15 ?

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-082.jpg?height=73&width=784&top_left_y=462&top_left_x=627)
(A) $A$
(B) $B$
(C) $C$
(D) $D$
(E) $E$
We label the points corresponding to the four larger dots as $W, X, Y$, and $Z$.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-156.jpg?height=108&width=786&top_left_y=2212&top_left_x=729)

Two of these four points must be multiples of 5 .

These multiples of 5 must differ from each other by a multiple of 5 .

We look at the possible differences between the four numbers $W, X, Y$, and $Z$ :

$$
X-W=1, Y-W=9, Y-X=8, Z-W=11, Z-X=10, Z-Y=2
$$

Only $X$ and $Z$ differ by a multiple of 5 . Thus, $X$ and $Z$ must be the two multiples of 5 .

We know that one of $A, B, C, D, E$ is a multiple of 5 , and so it must differ from $X$ by a multiple of 5 .

Since $D-X=5$, then $D$ is the only multiple of 5 among $A, B, C, D, E$ (the others differ from $X$ by $2,3,4$, and 6 .

Since $D$ is the only possibility that is a multiple of 5 , then $D$ must be the multiple of 15 .

(We can check that $X$ and $Z$ are multiples of 5 and $W$ and $Y$ are multiples of 3, which means that $D$ is also a multiple of 3 .)

ANSWER: (D)