Puzzle for October 13, 2021 ( )
Scratchpad
Find the 6-digit number ABCDE×F 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
Add F to both sides of eq.2: B + F + F = A – F + F which becomes eq.2a) B + 2×F = A In eq.5, replace A with B + 2×F (from eq.2a): B + C = B + 2×F + D Subtract B from both sides of the above equation: B + C – B = B + 2×F + D – B which becomes eq.5a) C = 2×F + D
Hint #2
Add E to both sides of eq.3: E + F + E = D – E + E which becomes eq.3a) 2×E + F = D In eq.5a, replace D with 2×E + F (from eq.3a): C = 2×F + 2×E + F which becomes eq.5b) C = 2×E + 3×F
Hint #3
In eq.4, substitute 2×E + 3×F for C (from eq.5b), and 2×E + F for D (from eq.3a): 2×E + 3×F + F = 2×E + F + E which becomes 2×E + 4×F = 3×E + F Subtract 2×E and F from both sides of the equation above: 2×E + 4×F – 2×E – F = 3×E + F – 2×E – F which makes 3×F = E
Hint #4
Substitute (3×F) for E into eq.5b: C = 2×(3×F) + 3×F which becomes C = 6×F + 3×F which makes C = 9×F
Hint #5
Substitute (3×F) for E into eq.3a: 2×(3×F) + F = D which becomes 6×F + F = D which makes 7×F = D
Hint #6
Substitute 7×F for D, 3×F for E, and 9×F for C in eq.6: 7×F + 3×F + F = A + 9×F – 7×F which becomes 11×F = A + 2×F Subtract 2×F from each side of the above equation: 11×F – 2×F = A + 2×F – 2×F which makes 9×F = A
Hint #7
Substitute 9×F for A in eq.2a: B + 2×F = 9×F Subtract 2×F from each side of the equation above: B + 2×F – 2×F = 9×F – 2×F which makes B = 7×F
Solution
Substitute 9×F for A and C, 7×F for B and D, and 3×F for E in eq.1: 9×F + 7×F + 9×F + 7×F + 3×F + F = 36 which simplifies to 36×F = 36 Divide both sides of the above equation by 36: 36×F ÷ 36 = 36 ÷ 36 which means F = 1 making A = C = 9×F = 9 × 1 = 9 B = D = 7×F = 7 × 1 = 7 E = 3×F = 3 × 1 = 3 and ABCDEF = 979731