Puzzle for February 8, 2022 ( )
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.4, replace C with D + E (from eq.2): B + D = D + E – B In the above equation, subtract D from both sides, and add B to both sides: B + D – D + B = D + E – B – D + B which simplifies to 2×B = E
Hint #2
In eq.6, replace E with 2×B: D + 2×B = A – 2×B Add 2×B to both sides of the above equation: D + 2×B + 2×B = A – 2×B + 2×B which becomes eq.6a) D + 4×B = A
Hint #3
In eq.3, substitute D + 4×B for A (from eq.6a), and 2×B for E: D + F = D + 4×B + 2×B which becomes D + F = D + 6×B Subtract D from each side of the equation above: D + F – D = D + 6×B – D which makes F = 6×B
Hint #4
Substitute 6×B for F in eq.5: 6×B – B = B + C which becomes 5×B = B + C Subtract B from each side of the above equation: 5×B – B = B + C – B which makes 4×B = C
Hint #5
Substitute 4×B for C, and 2×B for E in eq.2: 4×B = D + 2×B Subtract 2×B from both sides of the equation above: 4×B – 2×B = D + 2×B – 2×B which makes 2×B = D
Hint #6
Substitute 2×B for D in eq.6a: 2×B + 4×B = A which makes 6×B = A
Solution
Substitute 6×B for A and F, 4×B for C, and 2×B for D and E in eq.1: 6×B + B + 4×B + 2×B + 2×B + 6×B = 21 which simplifies to 21×B = 21 Divide both sides of the above equation by 21: 21×B ÷ 21 = 21 ÷ 21 which means B = 1 making A = F = 6×B = 6 × 1 = 6 C = 4×B = 4 × 1 = 4 D = E = 2×B = 2 × 1 = 2 and ABCDEF = 614226