Puzzle for October 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.5, replace F with B + E (from eq.4): B + B + E = C which becomes eq.5a) 2×B + E = C
Hint #2
In eq.2, replace C with 2×B + E (from eq.5a): B + 2×B + E = D which becomes eq.2a) 3×B + E = D
Hint #3
In eq.3, replace D with 3×B + E (from eq.2a): B + 3×B + E = A which becomes eq.3a) 4×B + E = A
Hint #4
In eq.6, substitute B + E for F (from eq.4), 4×B + E for A (from eq.3a), and 2×B + E for C (from eq.5a): E + B + E = 4×B + E + 2×B + E - E which becomes 2×E + B = 6×B + E Subtract B and E from both sides of the above equation: 2×E + B - B - E = 6×B + E - B - E which simplifies to E = 5×B
Hint #5
Substitute 5×B for E in eq.3a: 4×B + 5×B = A which makes 9×B = A
Hint #6
Substitute 5×B for E in eq.5a: 2×B + 5×B = C which makes 7×B = C
Hint #7
Substitute 5×B for E in eq.2a: 3×B + 5×B = D which makes 8×B = D
Hint #8
Substitute 5×B for E in eq.4: B + 5×B = F which makes 6×B = F
Solution
Substitute 9×B for A, 7×B for C, 8×B for D, 5×B for E, and 6×B for F in eq.1: 9×B + B + 7×B + 8×B + 5×B + 6×B = 36 which simplifies to 36×B = 36 Divide both sides of the above equation above by 36: 36×B ÷ 36 = 36 ÷ 36 which means B = 1 making A = 9×B = 9 × 1 = 9 C = 7×B = 7 × 1 = 7 D = 8×B = 8 × 1 = 8 E = 5×B = 5 × 1 = 5 F = 6×B = 6 × 1 = 6 and ABCDEF = 917856