Puzzle for May 8, 2023 ( )
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 B with C + D (from eq.4): D - C = C + D - D which becomes D - C = C Add C to both sides of the above equation: D - C + C = C + C which makes D = 2×C
Hint #2
In eq.4, substitute 2×C for D: B = C + 2×C which makes B = 3×C
Hint #3
In eq.3, substitute 3×C for B, and 2×C for D: F = 3×C + 2×C which makes F = 5×C
Hint #4
Substitute 5×C for F, and 2×C for D in eq.6: A - 5×C = 5×C - 2×C which becomes A - 5×C = 3×C Add 5×C to both sides of the equation above: A - 5×C + 5×C = 3×C + 5×C which makes A = 8×C
Hint #5
Substitute 8×C for A, and 5×C for F in eq.2: 8×C = E + 5×C Subtract 5×C from each side of the equation above: 8×C - 5×C = E + 5×C - 5×C which makes 3×C = E
Solution
Substitute 8×C for A, 3×C for B and E, 2×C for D, and 5×C for F in eq.1: 8×C + 3×C + C + 2×C + 3×C + 5×C = 22 which simplifies to 22×C = 22 Divide both sides of the above equation by 22: 22×C ÷ 22 = 22 ÷ 22 which means C = 1 making A = 8×C = 8 × 1 = 8 B = E = 3×C = 3 × 1 = 3 D = 2×C = 2 × 1 = 2 F = 5×C = 5 × 1 = 5 and ABCDEF = 831235