Puzzle for May 16, 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 E + F (from eq.3): E + F + E = B - E which becomes 2×E + F = B - E Add E to both sides of the above equation: 2×E + F + E = B - E + E which becomes eq.6a) 3×E + F = B
Hint #2
In eq.2, replace B with 3×E + F (from eq.6a): 3×E + F = D + F Subtract F from each side of the equation above: 3×E + F - F = D + F - F which makes 3×E = D
Hint #3
In eq.5, substitute 3×E for D: 3×E + E = F - E which becomes 4×E = F - E Add E to both sides of the equation above: 4×E + E = F - E + E which makes 5×E = F
Hint #4
Substitute 5×E for F in eq.6a: 3×E + 5×E = B which makes 8×E = B
Hint #5
Substitute 5×E for F in eq.3: C = E + 5×E which makes C = 6×E
Hint #6
Substitute 6×E for C, and 3×E for D in eq.4: A = 6×E + 3×E which makes A = 9×E
Solution
Substitute 9×E for A, 8×E for B, 6×E for C, 3×E for D, and 5×E for F in eq.1: 9×E + 8×E + 6×E + 3×E + E + 5×E = 32 which simplifies to 32×E = 32 Divide both sides of the above equation above by 32: 32×E ÷ 32 = 32 ÷ 32 which means E = 1 making A = 9×E = 9 × 1 = 9 B = 8×E = 8 × 1 = 8 C = 6×E = 6 × 1 = 6 D = 3×E = 3 × 1 = 3 F = 5×E = 5 × 1 = 5 and ABCDEF = 986315