Puzzle for September 4, 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 F with B + E (from eq.3): A + E = B + E - A In the above equation, subtract E from both sides, and add A to both sides: A + E - E + A = B + E - A - E + A which makes 2×A = B
Hint #2
In eq.6, replace B with 2×A, and C with D + F (from eq.4): 2×A + D = D + F - 2×A In the above equation, subtract D from both sides, and add 2×A to both sides: 2×A + D - D + 2×A = D + F - 2×A - D + 2×A which makes 4×A = F
Hint #3
In eq.2, substitute 4×A for F in eq.2: D = A + 4×A which makes D = 5×A
Hint #4
Substitute 5×A for D, and 4×A for F in eq.4: C = 5×A + 4×A which makes C = 9×A
Hint #5
Substitute 4×A for F, and 2×A for B in eq.3: 4×A = 2×A + E Subtract 2×A from each side of the equation above: 4×A - 2×A = 2×A + E - 2×A which makes 2×A = E
Solution
Substitute 2×A for B and E, 9×A for C, 5×A for D, and 4×A for F in eq.1: A + 2×A + 9×A + 5×A + 2×A + 4×A = 23 which simplifies to 23×A = 23 Divide both sides of the above equation by 23: 23×A ÷ 23 = 23 ÷ 23 which means A = 1 making B = E = 2×A = 2 × 1 = 2 C = 9×A = 9 × 1 = 9 D = 5×A = 5 × 1 = 5 F = 4×A = 4 × 1 = 4 and ABCDEF = 129524