Puzzle for November 22, 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.5, replace C with A + E + F (from eq.4): A + E – F = A + E + F – E which becomes A + E – F = A + F In the above equation, subtract A from both sides, and add F to both sides: A + E – F – A + F = A + F – A + F which makes E = 2×F
Hint #2
In eq.3, replace E with 2×F: 2×F + F = A – F which becomes 3×F = A – F Add F to both sides of the above equation: 3×F + F = A – F + F which makes 4×F = A
Hint #3
In eq.2, substitute 4×F for A, and 2×F for E: 4×F + 2×F = D which makes 6×F = D
Hint #4
Substitute 4×F for A, and 2×F for E in eq.4: 4×F + 2×F + F = C which makes 7×F = C
Hint #5
Substitute 6×F for D, and 2×F for E in eq.6: 6×F + 2×F + F = B which makes 9×F = B
Solution
Substitute 4×F for A, 9×F for B, 7×F for C, 6×F for D, and 2×F for E in eq.1: 4×F + 9×F + 7×F + 6×F + 2×F + F = 29 which simplifies to 29×F = 29 Divide both sides of the above equation by 29: 29×F ÷ 29 = 29 ÷ 29 which means F = 1 making A = 4×F = 4 × 1 = 4 B = 9×F = 9 × 1 = 9 C = 7×F = 7 × 1 = 7 D = 6×F = 6 × 1 = 6 E = 2×F = 2 × 1 = 2 and ABCDEF = 497621