Puzzle for July 3, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* AB and CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
Add C to both sides of eq.4: E + C = D - C + C which becomes E + C = D In eq.3, replace D with E + C: E + C + E + F = C Subtract C from both sides of the above equation: E + C + E + F - C = C - C which becomes 2×E + F = 0 Since E and F are non-negative integers, the equation above makes E = 0 and F = 0
Hint #2
In eq.3, replace both E and F with 0: D + 0 + 0 = C which makes D = C
Hint #3
In eq.2, substitute C for D: B = C + C which means B = 2×C
Hint #4
eq.5 may be written as: 10×C + D + A + C + F = 10×A + B - A which becomes 11×C + D + A + F = 9×A + B Substitute C for D, 0 for F, and 2×C for B in the above equation: 11×C + C + A + 0 = 9×A + 2×C which becomes 12×C + A = 9×A + 2×C Subtract A and 2×C from both sides: 12×C + A - A - 2×C = 9×A + 2×C - A - 2×C which becomes 10×C = 8×A Divide both sides by 8: 10×C ÷ 8 = 8×A ÷ 8 which makes 1¼×C = A
Solution
Substitute 1¼×C for A, 2×C for B, C for D, and 0 for E and F in eq.1: 1¼×C + 2×C + C + C + 0 + 0 = 21 which simplifies to 5¼×C = 21 Divide both sides of the equation above by 5¼: 5¼×C ÷ 5¼ = 21 ÷ 5¼ which means C = 4 making A = 1¼×C = 1¼ × 4 = 5 B = 2×C = 2×4 = 8 D = C = 4 and ABCDEF = 584400