Puzzle for May 23, 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.6, replace E + F with B + C (from eq.5): A + D = B + B + C which becomes eq.6a) A + D = 2×B + C
Hint #2
In eq.6a, replace C with B + D (from eq.4): A + D = 2×B + B + D which becomes A + D = 3×B + D Subtract D from each side of the above equation: A + D - D = 3×B + D - D which makes A = 3×B
Hint #3
In eq.3, substitute 3×B for A: eq.3a) D = 3×B + E
Hint #4
Substitute 3×B for A, and 3×B + E for D (from eq.3a) into eq.6: 3×B + 3×B + E = B + E + F which becomes 6×B + E = B + E + F Subtract E and B from each side of the equation above: 6×B + E - E - B = B + E + F - E - B which simplifies to 5×B = F
Hint #5
Substitute 3×B for A, and 5×B for F in eq.2: C = 3×B + 5×B which makes C = 8×B
Hint #6
Substitute 8×B for C in eq.4: 8×B = B + D Subtract B from both sides of the above equation: 8×B - B = B + D - B which makes 7×B = D
Hint #7
Substitute 7×B for D, and 3×B for A in eq.3: 7×B = 3×B + E Subtract 3×B from both sides of the equation above: 7×B - 3×B = 3×B + E - 3×B which makes 4×B = E
Solution
Substitute 3×B for A, 8×B for C, 7×B for D, 4×B for E, and 5×B for F in eq.1: 3×B + B + 8×B + 7×B + 4×B + 5×B = 28 which simplifies to 28×B = 28 Divide both sides of the above equation by 28: 28×B ÷ 28 = 28 ÷ 28 which means B = 1 making A = 3×B = 3 × 1 = 3 C = 8×B = 8 × 1 = 8 D = 7×B = 7 × 1 = 7 E = 4×B = 4 × 1 = 4 F = 5×B = 5 × 1 = 5 and ABCDEF = 318745