Question

A number is called "bright" if it is 34 times larger than the sum of its digits. How many "bright" three-digit numbers are there?

A : 0

B : 1

C : 2

D : 3

E : 4

Answer 1

Explanation:

Let the number be
`\overline{abc}=100a+10b+c`.

`100a+10b+c=34a+34b+34c \\ \\ 66a-24b-33c=0 \\ \\ 22a-8b-11c=0`

Taking the equation mod
`11` yields
`-8b \equiv \pmod{11}`, meaning
`b=0`.

So,
`22a-11c=0 \implies c=2a`.

So, the only possibilites are
`(a,c)=(1,2), (2,4),(3,6), (4,8)`.

Therefore, there are
`4` such numbers.