Short answer: The two digit number is 36.
Long answer:
7 * x1x2 = 4 * x2x1
"If the difference between the number is 3. . ."
Last digits:
x2: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
7*x2: 0, 7, 4, 1, 8, 5, 2, 9, 6, 1
x1: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
4*x1: 0, 4, 8, 2, 6, 0, 4, 8, 2, 6
The only way this works is when these last digits for 7*x2 and 4*x1 are the same. That means the 7*x2 line can only be:
7*x2: 0, 4, 8, 2, 6
Which means the possible values for x2 are:
x2: 0, 2, 4, 6, 8
Now let's look at x1. Right now the possible values for x1 are:
x1: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
But we have to end up with a choice from x1 and x2 having a difference of 3 (odd number). There is no way to get an odd number by subtracting an even number from an even number. That means x1 has to be odd. Our possible values for x1 are now:
x1: 1, 3, 5, 7, 9
And our possible values for x2 are:
x2: 0, 2, 4, 6, 8
The possible combinations for x1x2 and x2x1 are:
10, 01
12, 21
14, 41
16, 61
18, 81
30, 03
32, 23
34, 43
36, 63
38, 83
50, 05
52, 25
54, 45
56, 65
58, 85
70, 07
72, 27
74, 47
76, 67
78, 87
90, 09
92, 29
94, 49
96, 69
98, 89
We want 7*x1x2 = 4*x2x1, so we can do 7 * the first column and 4 * the second column:
70, 4
84, 84
98, 164
112, 244
126, 324
210, 12
224, 92
238, 172
252, 252
266, 332
350, 20
364, 100
378, 180
392, 260
406, 340
490, 28
504, 108
518, 188
532, 268
546, 348
630, 36
644, 116
658, 196
672, 276
686, 356
But since we want 7*x1x2 to equal 4*x2x1, that list reduces to:
84, 84
252, 252
The corresponding x1 and x2 values are:
12, 21
36, 63
But the difference between x1 and x2 is 3, so we can't use x1 = 1, x2=2. We have to use x1 = 3, x2 = 6.
Answer: The two digit number is 36.
Check: 7 * 36 = 4 * 63
252 = 252 good.
6 - 3 = 3 good.
