Suppose i have two two-digit numbers, and I write them one after another (with the larger two-digit number first) to give a four-digit number.
From this four-digit number, I subtract the difference between the two two-digit numbers.
The result is 5689.
What is the smaller of the two numbers?
Let A = the larger 2-digit number
Let B = the smaller 2-digit number
The 4-digit number = 100A+B
The difference between A and B is A-B
Subtract that from the 4-digit number and that must equal 5689
100A+B-(A-B) = 5689
100A+B-A+B = 5689
99A+2B = 5689
The smallest coefficient in absolute value is 2.
Write 99 and 5689 in terms of their nearest multiple
of 2 which does not exceed them:
(98 + 1)A + 2B = 5688 + 1
98A + A + 2B = 5688 + 1
Divide thru by 2
49A + A/2 + B = 2844 + 1/2
Isolate the fractions:
A/2 - 1/2 = 2844 - 49A - B
The right side is an integer, so the
left side is also an integer. Let
that integer be C. Set each side = C:
A/2 - 1/2 = C; 2844 - 49A - B = C
A - 1 = 2C
A = 2C + 1
Substitute in
2844 - 49A - B = C
2844 - 49(2C + 1) - B = C
2844 - 98C - 49 - B = C
2795 - 99C - B = 0
2795 - 99C = B
Since B is a 2-digit number
9 < B < 100
9 < 2795 - 99C < 100
Solve for C in the middle,
add -2795 to all 3 sides:
-2785 < -99C < -2695
Divide all three sides by -99,
reversing the inequalities:
__ _
28.14 > C > 27.2 (the bars indicate repeating decimals)
C is an integer and there is only one integer
between those values. Thus C = 28.
Then B = 2795 - 99C = 2795 - 99(28) = 2795 - 2772 = 23
and A = 2C + 1 = 2(28) + 1 = 56 + 1 = 57
Answer: A = 57, B = 23
The larger 2-digit number is 57, the smaller is 23
Checking:
57 - 23 = 34
5723 - 34 = 5689
So that is correct.
Edwin