Puzzle for June 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
Subtract the left and right sides of eq.5 from the left and right sides of eq.4, respectively: A + E – (A + F) = C + F – (C – F) which becomes A + E – A – F = C + F – C + F which becomes E – F = 2×F Add F to both sides of the above equation: E – F + F = 2×F + F which makes E = 3×F
Hint #2
In eq.6, replace A + D with C + E (from eq.3): C + E + E = B + C which becomes C + 2×E = B + C Subtract C from both sides of the equation above: C + 2×E – C = B + C – C which makes eq.6a) 2×E = B
Hint #3
In eq.6a, substitute (3×F) for E: 2×(3×F) = B which makes 6×F = B
Hint #4
Substitute 6×F for B in eq.2: A = 6×F + F which makes A = 7×F
Hint #5
Substitute 7×F for A in eq.5: 7×F + F = C – F which becomes 8×F = C – F Add F to both sides of the above equation: 8×F + F = C – F + F which makes 9×F = C
Hint #6
Substitute 7×F for A, 9×F for C, and 3×F for E in eq.3: 7×F + D = 9×F + 3×F which becomes 7×F + D = 12×F Subtract 7×F from each side of the equation above: 7×F + D – 7×F = 12×F – 7×F which makes D = 5×F
Solution
Substitute 7×F for A, 6×F for B, 9×F for C, 5×F for D, and 3×F for E in eq.1: 7×F + 6×F + 9×F + 5×F + 3×F + F = 31 which simplifies to 31×F = 31 Divide both sides of the above equation by 31: 31×F ÷ 31 = 31 ÷ 31 which means F = 1 making A = 7×F = 7 × 1 = 7 B = 6×F = 6 × 1 = 6 C = 9×F = 9 × 1 = 9 D = 5×F = 5 × 1 = 5 E = 3×F = 3 × 1 = 3 and ABCDEF = 769531