Puzzle for November 9, 2020 ( )
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.2: B + F + F = C – F + F which becomes eq.2a) B + 2×F = C In eq.3, replace C with B + 2×F (from eq.2a): B + 2×F – B = D which makes 2×F = D
Hint #2
In eq.6, replace C with B + 2×F (from eq.2a), and D with 2×F: B + E = B + 2×F + 2×F + F which becomes B + E = B + 5×F Subtract B from each side of the above equation: B + E – B = B + 5×F – B which makes E = 5×F
Hint #3
In eq.5, substitute 5×F for E, and 2×F for D: 5×F – F = A + 2×F which becomes 4×F = A + 2×F Subtract 2×F from each side of the equation above: 4×F – 2×F = A + 2×F – 2×F which makes 2×F = A
Hint #4
Substitute 2×F for D, 5×F for E, and 2×F for A in eq.4: 2×F + 5×F = 2×F + B which becomes 7×F = 2×F + B Subtract 2×F from each side of the equation above: 7×F – 2×F = 2×F + B – 2×F which makes 5×F = B
Hint #5
Substitute 5×F for B, and 2×F for D in eq.3: C – 5×F = 2×F Add 5×F to both sides of the equation above: C – 5×F + 5×F = 2×F + 5×F which makes C = 7×F
Solution
Substitute 2×F for A and D, 5×F for B and E, and 7×F for C in eq.1: 2×F + 5×F + 7×F + 2×F + 5×F + F = 22 which simplifies to 22×F = 22 Divide both sides of the equation above by 22: 22×F ÷ 22 = 22 ÷ 22 which means F = 1 making A = D = 2×F = 2 × 1 = 2 B = E = 5×F = 5 × 1 = 5 C = 7×F = 7 × 1 = 7 and ABCDEF = 257251