Puzzle for July 31, 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 C with B + F (from eq.3): E + F = B + F - E In the equation above, subtract F from both sides, and add E to both sides: E + F - F + E = B + F - E - F + E which makes 2×E = B
Hint #2
In eq.2, replace B with 2×E: A = 2×E + E which makes A = 3×E
Hint #3
In eq.5, substitute 3×E for A: D - 3×E = 3×E - E which becomes D - 3×E = 2×E Add 3×E to both sides of the above equation: D - 3×E + 3×E = 2×E + 3×E which makes D = 5×E
Hint #4
Substitute 3×E for A, and 5×E for D in eq.4: C = 3×E + 5×E which makes C = 8×E
Hint #5
Substitute 8×E for C, and 2×E for B in eq.3: 8×E = 2×E + F Subtract 2×E from each side of the equation above: 8×E - 2×E = 2×E + F - 2×E which makes 6×E = F
Solution
Substitute 3×E for A, 2×E for B, 8×E for C, 5×E for D, and 6×E for F in eq.1: 3×E + 2×E + 8×E + 5×E + E + 6×E = 25 which simplifies to 25×E = 25 Divide both sides of the above equation by 25: 25×E ÷ 25 = 25 ÷ 25 which means E = 1 making A = 3×E = 3 × 1 = 3 B = 2×E = 2 × 1 = 2 C = 8×E = 8 × 1 = 8 D = 5×E = 5 × 1 = 5 F = 6×E = 6 × 1 = 6 and ABCDEF = 328516