Puzzle for June 18, 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.
* CD is a 2-digit number (not C×D).
Scratchpad
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Hint #1
eq.4 may be written as: A + C + D = 10×C + D Subtract both C and D from each side of the equation above: A + C + D - C - D = 10×C + D - C - D which simplifies to A = 9×C
Hint #2
In eq.3, replace A with 9×C: 10×C + D - C = 9×C + B - D which becomes 9×C + D = 9×C + B - D In the above equation, add D to both sides, and subtract 9×C from each side: 9×C + D + D - 9×C = 9×C + B - D + D - 9×C which simplifies to eq.3a) 2×D = B
Hint #3
In eq.2, replace B with 2×D: 2×D + D = A which makes 3×D = A Substitute 9×C for A in the above equation: 3×D = 9×C Divide both sides by 3: 3×D ÷ 3 = 9×C ÷ 3 which makes D = 3×C
Hint #4
Substitute (3×C) for D in eq.3a: 2×(3×C) = B which makes 6×C = B
Solution
Substitute 9×C for A, 6×C for B, and 3×C for D in eq.1: 9×C + 6×C + C + 3×C = 19 which simplifies to 19×C = 19 Divide both sides of the above equation by 19: 19×C ÷ 19 = 19 ÷ 19 which means C = 1 making A = 9×C = 9 × 1 = 9 B = 6×C = 6 × 1 = 6 D = 3×C = 3 × 1 = 3 and ABCD = 9613