Puzzle for July 26, 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.6, replace B with A + D (from eq.2): A + E = A + D – E In the above equation, subtract A from both sides, and add E to both sides: A + E – A + E = A + D – E – A + E which makes 2×E = D
Hint #2
In eq.5, replace B – F with D + E (from eq.3): F – D = D + E Add D to both sides of the equation above: F – D + D = D + E + D which becomes eq.5a) F = 2×D + E
Hint #3
In eq.5a, substitute (2×E) for D: F = 2×(2×E) + E which becomes F = 4×E + E which makes F = 5×E
Hint #4
Substitute 2×E for D, and 5×E for F into eq.3: 2×E + E = B – 5×E which becomes 3×E = B – 5×E Add 5×E to both sides of the above equation: 3×E + 5×E = B – 5×E + 5×E which makes 8×E = B
Hint #5
Substitute 8×E for B in eq.6: A + E = 8×E – E which becomes A + E = 7×E Subtract E from each side of the above equation: A + E – E = 7×E – E which makes A = 6×E
Hint #6
Substitute 5×E for F, and 6×E for A in eq.4: E + 5×E = 6×E + C which becomes 6×E = 6×E + C Subtract 6×E from each side of the equation above: 6×E – 6×E = 6×E + C – 6×E which makes 0 = C
Solution
Substitute 6×E for A, 8×E for B, 0 for C, 2×E for D, and 5×E for F in eq.1: 6×E + 8×E + 0 + 2×E + E + 5×E = 22 which simplifies to 22×E = 22 Divide both sides of the above equation by 22: 22×E ÷ 22 = 22 ÷ 22 which means E = 1 making A = 6×E = 6 × 1 = 6 B = 8×E = 8 × 1 = 8 D = 2×E = 2 × 1 = 2 F = 5×E = 5 × 1 = 5 and ABCDEF = 680215