Puzzle for July 14, 2021 ( )
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.3, replace C with E + F (from eq.2): E + F + F = B + E which becomes E + 2×F = B + E Subtract E from each side of the above equation: E + 2×F – E = B + E – E which makes eq.3a) 2×F = B
Hint #2
In eq.6, replace B with 2×F (from eq.3a): D + F = 2×F – D – F which becomes D + F = F – D In the equation above, subtract F from both sides, and add D to both sides: D + F – F + D = F – D – F + D which makes 2×D = 0 which means D = 0
Hint #3
In eq.5, substitute 0 for D: A + 0 – E = C + E which becomes A – E = C + E Add E to both sides of the above equation: A – E + E = C + E + E which becomes eq.5a) A = C + 2×E
Hint #4
In eq.4, substitute 2×F for B (from eq.3a), C + 2×E for A (from eq.5a), and 0 for D: 2×F + F = C + 2×E + C + 0 which becomes eq.4a) 3×F = 2×C + 2×E
Hint #5
Substitute (E + F) for C (from eq.2) in eq.4a: 3×F = 2×(E + F) + 2×E which is equivalent to 3×F = 2×E + 2×F + 2×E which becomes 3×F = 4×E + 2×F Subtract 2×F from both sides of the equation above: 3×F – 2×F = 4×E + 2×F – 2×F which makes F = 4×E
Hint #6
Substitute 4×E for F in eq.2: E + 4×E = C which makes 5×E = C
Hint #7
Substitute 5×E for C in eq.5a: A = 5×E + 2×E which makes A = 7×E
Hint #8
Substitute (4×E) for F in eq.3a: 2×(4×E) = B which makes 8×E = B
Solution
Substitute 7×E for A, 8×E for B, 5×E for C, 0 for D, and 4×E for F in eq.1: 7×E + 8×E + 5×E + 0 + E + 4×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 = 7×E = 7 × 1 = 7 B = 8×E = 8 × 1 = 8 C = 5×E = 5 × 1 = 5 F = 4×E = 4 × 1 = 4 and ABCDEF = 785014