Puzzle for October 7, 2019 ( )
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.2, replace D with B + F (from eq.3): E + F = B + B + F which becomes E + F = 2×B + F Subract F from both sides of the equation above: E + F – F = 2×B + F – F which makes E = 2×B
Hint #2
In eq.5, replace E with 2×B: B + 2×B = A which makes 3×B = A
Hint #3
In eq.4, substitute 3×B for A: C = 3×B + B which makes C = 4×B
Hint #4
Substitute 3×B for A, 4×B for C, and 2×B for E in eq.6: 3×B + B + 4×B = D + 2×B which becomes 8×B = D + 2×B Subtract 2×B from both sides of the above equation: 8×B – 2×B = D + 2×B – 2×B which makes 6×B = D
Hint #5
Substitute 6×B for D in eq.3: 6×B = B + F Subtract B from both sides of the above equation: 6×B – B = B + F – B which makes 5×B = F
Solution
Substitute 3×B for A, 4×B for C, 6×B for D, 2×B for E, and 5×B for F in eq.1: 3×B + B + 4×B + 6×B + 2×B + 5×B = 21 which simplifies to 21×B = 21 Divide both sides of the equation above by 21: 21×B ÷ 21 = 21 ÷ 21 which means B = 1 making A = 3×B = 3 × 1 = 3 C = 4×B = 4 × 1 = 4 D = 6×B = 6 × 1 = 6 E = 2×B = 2 × 1 = 2 F = 5×B = 5 × 1 = 5 and ABCDEF = 314625