Puzzle for September 3, 2019 ( )
Scratchpad
Find the 4-digit number ABCD by solving the following equations:
A, B, C, and D each represent a one-digit non-negative integer.
* AB and BC are 2-digit numbers (not A×B or B×C).
Scratchpad
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Hint #1
In eq.3, replace D with A + C (from eq.2): B + C = A + A + C Subtract C from each side of the above equation: B + C – C = A + A + C – C which makes B = 2×A
Hint #2
eq.4 may be written as: 10×A + B + C + D = A + 10×B + C Subtract A, B, and C from each side of the above equation: 10×A + B + C + D – A – B – C = A + 10×B + C – A – B – C which simplifies to 9×A + D = 9×B Replace B with (2×A): 9×A + D = 9×(2×A) which is equivalent to 9×A + D = 18×A Subtract 9×A from both sides: 9×A + D – 9×A = 18×A – 9×A which makes D = 9×A
Hint #3
In eq.2, substitute 9×A for D: 9×A = A + C Subtract A from each side of the above equation: 9×A – A = A + C – A which means 8×A = C
Solution
Substitute 2×A for B, 8×A for C, and 9×A for D in eq.1: A + 2×A + 8×A + 9×A = 20×A which simplifies to 20×A = 20 Divide both sides by 20: 20×A ÷ 20 = 20 ÷ 20 which means A = 1 making B = 2×A = 2 × 1 = 2 C = 8×A = 8 × 1 = 8 D = 9×A = 9 × 1 = 9 and ABCD = 1289