Puzzle for January 13, 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
In eq.5, replace A with C + D (from eq.4): C + E = C + D + D which becomes C + E = C + 2×D Subtract C from each side of the equation above: C + E – C = C + 2×D – C which makes eq.5a) E = 2×D
Hint #2
In eq.6, replace E with 2×D: B + D = 2×D – B In the above equation, subtract D from both sides, and add B to each side: B + D – D + B = 2×D – B – D + B which makes 2×B = D
Hint #3
In eq.5a, substitute (2×B) for D: E = 2×(2×B) which makes E = 4×B
Hint #4
Substitute 2×B for D, and 4×B for E in eq.3: B + C = 2×B + 4×B which becomes B + C = 6×B Subtract B from both sides of the equation above: B + C – B = 6×B – B which makes C = 5×B
Hint #5
Substitute 5×B for C, and 2×B for D in eq.4: A = 5×B + 2×B which makes A = 7×B
Hint #6
Substitute 4×B for E, 7×B for A, and 5×B for C in eq.2: 4×B + F = 7×B + 5×B which becomes 4×B + F = 12×B Subtract 4×B from each side of the above equation: 4×B + F – 4×B = 12×B – 4×B which makes F = 8×B
Solution
Substitute 7×B for A, 5×B for C, 2×B for D, 4×B for E, and 8×B for F in eq.1: 7×B + B + 5×B + 2×B + 4×B + 8×B = 27 which simplifies to 27×B = 27 Divide both sides of the equation above by 27: 27×B ÷ 27 = 27 ÷ 27 which means B = 1 making A = 7×B = 7 × 1 = 7 C = 5×B = 5 × 1 = 5 D = 2×B = 2 × 1 = 2 E = 4×B = 4 × 1 = 4 F = 8×B = 8 × 1 = 8 and ABCDEF = 715248