Puzzle for August 29, 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.6, replace D with E + F (from eq.2): E + F – F = F – E which becomes E = F – E Add E to both sides of the above equation: E + E = F – E + E which makes 2×E = F
Hint #2
In eq.2, replace F with 2×E: D = E + 2×E which makes D = 3×E
Hint #3
In eq.3, substitute 3×E for D: A = 3×E + E which makes A = 4×E
Hint #4
Substitute 2×E for F, and 4×E for A in eq.4: 2×E – B = 4×E – 2×E which becomes 2×E – B = 2×E Subtract 2×E from each side of the equation above: 2×E – B – 2×E = 2×E – 2×E which makes –B = 0 which means B = 0
Hint #5
Substitute 4×E for A, and 0 for B in eq.5: C – 4×E = 4×E – 0 which becomes C – 4×E = 4×E Add 4×E to both sides of the equation above: C – 4×E + 4×E = 4×E + 4×E which makes C = 8×E
Solution
Substitute 4×E for A, 0 for B, 8×E for C, 3×E for D, and 2×E for F in eq.1: 4×E + 0 + 8×E + 3×E + E + 2×E = 18 which simplifies to 18×E = 18 Divide both sides of the above equation by 18: 18×E ÷ 18 = 18 ÷ 18 which means E = 1 making A = 4×E = 4 × 1 = 4 C = 8×E = 8 × 1 = 8 D = 3×E = 3 × 1 = 3 F = 2×E = 2 × 1 = 2 and ABCDEF = 408312