Please help with this problem!

Question

What is the positive three-digit integer that is four times the product of its digits?

Answer 1

I am mostly wondering if there's an equation to solve this problem...I have worked this problem with the guess, check, and revise method for about half an hour, and cannot figure it out. I tried writing an equation, but it didn't work out. So could any of you figure out an equation? You don't have to tell me the answer, just help me find an equation.

Thanks!!

Answer 2

My equation I tried to use is ABC=4(AxBxC)

I tried to solve it but whatever I did, I wouldn't get anywhere. Does anyone have any suggestions to make an equation that will work?

Answer 3

If A, B, and C are the digits of the number, then the number would be
100A + 10B + C

so your equation would be

100A + 10B + C = 4ABC
100A + 10B = 4ABC - C
100A + 10B = C(4AB - 1)
C = (100A + 10B)/(4AB - 1)
= 10(10A + B)/(4AB - 1)

now a bit of logic.
you know the numerator ends in a zero, so the bottom has to be divisible by 10 or 5
something multiplied by 4 can only end in 4,8,2,6, or 0 so after subtracting the 1 the denominator can only end in
3,7,1,5, or 9 , never in 10

also neither A nor B can be zeros, since then the denominator would be -1

also A and B can only be between 1 and 9

so we are looking for A and B which makes 4AB - 1 end in a 5.

After a few 'guesses' I found A=3 and B=8 gives me C=4

so the number is 384

check 4(3x8x4) = 384