Puzzle for April 26, 2021 ( )
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.
Scratchpad
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Hint #1
In eq.6, replace E with C + F (from eq.3): C + D = C + F – D In the equation above, subtract C from both sides, and add D to both sides: C + D – C + D = C + F – D – C + D which simplifies to 2×D = F
Hint #2
In eq.2, replace F with 2×D: D + 2×D = C which makes 3×D = C
Hint #3
In eq.3, substitute 3×D for C, and 2×D for F: E = 3×D + 2×D which makes E = 5×D
Hint #4
Substitute 3×D for C, and 5×D for E in eq.5: A + D = 3×D + 5×D which becomes A + D = 8×D Subtract D from both sides of the equation above: A + D – D = 8×D – D which makes A = 7×D
Hint #5
Substitute 2×D for F, and 7×D for A in eq.4: B + 2×D = 7×D Subtract 2×D from each side of the above equation: B + 2×D – 2×D = 7×D – 2×D which makes B = 5×D
Solution
Substitute 7×D for A, 5×D for B and E, 3×D for C, and 2×D for F in eq.1: 7×D + 5×D + 3×D + D + 5×D + 2×D = 23 which simplifies to 23×D = 23 Divide both sides of the above equation by 23: 23×D ÷ 23 = 23 ÷ 23 which means D = 1 making A = 7×D = 7×1 = 7 B = E = 5×D = 5×1 = 5 C = 3×D = 3×1 = 3 F = 2×D = 2×1 = 2 and ABCDEF = 753152