Puzzle for May 2, 2022 ( )
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.5, replace F with D + E (from eq.2): E – D = D + E – E which becomes E – D = D Add D to both sides of the above equation: E – D + D = D + D which makes E = 2×D
Hint #2
In eq.2, replace E with 2×D: F = D + 2×D which makes F = 3×D
Hint #3
In eq.4, substitute 2×D for E, and 3×D for F: A – 2×D = D + 3×D which becomes A – 2×D = 4×D Add 2×D to both sides of the above equation: A – 2×D + 2×D = 4×D + 2×D which makes A = 6×D
Hint #4
Substitute 6×D for A in eq.3: C = 6×D + D which makes C = 7×D
Hint #5
Substitute 6×D for A, and 2×D for E in eq.6: B + D = 6×D + 2×D which becomes B + D = 8×D Subtract D from both sides of the equation above: B + D – D = 8×D – D which makes B = 7×D
Solution
Substitute 6×D for A, 7×D for B and C, 2×D for E, and 3×D for F in eq.1: 6×D + 7×D + 7×D + D + 2×D + 3×D = 26 which simplifies to 26×D = 26 Divide both sides of the above equation by 26: 26×D ÷ 26 = 26 ÷ 26 which means D = 1 making A = 6×D = 6 × 1 = 6 B = C = 7×D = 7 × 1 = 7 E = 2×D = 2 × 1 = 2 F = 3×D = 3 × 1 = 3 and ABCDEF = 677123