6. A box contains red, white, and blue balls. The number of red balls is an even number, and the total number of balls in the box is less than 100. The number of white and blue balls together is 4 times the number of red balls. The number of red and blue balls together is 6 times the number of white balls.

How many balls are in the box?
A) 28
B) 30
C) 35
D) 70
E) 84
A6. D) $70 \quad$ Suppose there are $R$ red balls, $B$ blue balls, and $W$ white balls in the container. From the problem, we have that

$$
W+B=4 R \text { and } R+B=6 W \text {. }
$$

There are various ways to solve this further; here we provide one. If we subtract one equation from the other, we find that $W-R=4 R-6 W$, or $7 W=5 R$. Since 7 is a prime number and 5 is not divisible by 7, $R$ must be divisible by 7. Moreover, $R$ must also be even, so $R$ is a multiple of 14. If we take $R=14$, we get $W=\frac{5}{7} R=10$ and $B=4 R-W=46$ successively. If we take $R$ greater than 14, then $R$ is at least 28, and we find that $W$ is at least 20 and $B$ is at least 92. However, it is given that the total number of balls is less than 100, so this is not possible, and $R$ must be 14. In total, there are then $14+10+46=70$ balls in the container.