# The number game

There is a number,
the second digit of which is smaller than the first digit by $$4$$, and if the number was divided by the digits' sum, the quotient would be $$7$$.

What is this number.

• The hint it leaves only 6 possible numbers. That is superfluous imho. Commented Jul 26, 2020 at 11:21
• @FlorianF Please tell me a better hint then. I didn't find one. Commented Jul 26, 2020 at 13:43
• I would say the problem is already easy enough. No hint needed. Commented Jul 26, 2020 at 13:55
• @FlorianF Ok. edited. Commented Jul 27, 2020 at 7:47
• I think it was better to mention that the number has two digits in total.
– user70797
Commented Aug 6, 2020 at 21:19

For number AB:

1. A - B = 4
2. (10A + B) ÷ (A + B) = 7

Rearrange 1. to A = B + 4

Substitute A into 2. -> (10[B + 4] + B) ÷ ([B + 4] + B) = 7

Simplify 2.
- (10B + 40 + B) ÷ (2B + 4) = 7
- 11B + 40 = 14B + 28
- 12 = 3B
- B = 4

A = B + 4 = 8

Number is 84

• @math Thanks. How do you use the spoiler feature?
– asg
Commented Jul 26, 2020 at 22:46
• Commented Jul 27, 2020 at 7:44

After some guess and check, the answer is,

84

Since

8 - 4 = 4
84 / 12 = 7

• Assuming that "second digit" and "first digit" are counted from left, it is actually possible to prove that this is the unique answer, even without the hint. Commented Jul 25, 2020 at 18:50
• It’s an algebra problem. The answer added by @asg gives the steps. Commented Jul 26, 2020 at 5:51