Puzzle for March 1, 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
Add F to both sides of eq.6: E – F + F = C + F + F which becomes eq.6a) E = C + 2×F In eq.4, replace E with C + 2×F (from eq.6a): C + D = C + 2×F Subtract C from both sides of the above equation: C + D – C = C + 2×F – C which makes D = 2×F
Hint #2
In eq.2, replace D with 2×F, and E with C + 2×F (from eq.6a): 2×F + C + 2×F + F = A + C which becomes C + 5×F = A + C Subtract C from each side of the equation above: C + 5×F – C = A + C – C which makes 5×F = A
Hint #3
In eq.3, substitute 5×F for A: E = 5×F + F which makes E = 6×F
Hint #4
Substitute 6×F for E in eq.6a: 6×F = C + 2×F Subtract 2×F from both sides of the equation above: 6×F – 2×F = C + 2×F – 2×F which makes 4×F = C
Hint #5
Substitute 5×F for A, 2×F for D, and 6×F for E in eq.5: 5×F + 2×F = B + 6×F + F which becomes 7×F = B + 7×F Subtract 7×F from both sides of the above equation: 7×F – 7×F = B + 7×F – 7×F which makes 0 = B
Solution
Substitute 5×F for A, 0 for B, 4×F for C, 2×F for D, and 6×F for E in eq.1: 5×F + 0 + 4×F + 2×F + 6×F + F = 18 which simplifies to 18×F = 18 Divide both sides of the above equation by 18: 18×F ÷ 18 = 18 ÷ 18 which means F = 1 making A = 5×F = 5 × 1 = 5 C = 4×F = 4 × 1 = 4 D = 2×F = 2 × 1 = 2 E = 6×F = 6 × 1 = 6 and ABCDEF = 504261