# 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. Jul 26 '20 at 11:21
• @FlorianF Please tell me a better hint then. I didn't find one.
– math
Jul 26 '20 at 13:43
• I would say the problem is already easy enough. No hint needed. Jul 26 '20 at 13:55
• @FlorianF Ok. edited.
– math
Jul 27 '20 at 7:47
• I think it was better to mention that the number has two digits in total. Aug 6 '20 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
Jul 26 '20 at 22:46
• – math
Jul 27 '20 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. Jul 25 '20 at 18:50
• It’s an algebra problem. The answer added by @asg gives the steps. Jul 26 '20 at 5:51