Puzzle for November 3, 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
In eq.3, replace A + B with C – B (from eq.5): E = C – B + F Add B to both sides of the above equation: E + B = C – B + F + B which becomes eq.3a) E + B = C + F
Hint #2
In eq.4, substitute E + B for C + F (from eq.3a): B + D = E + B Subtract B from each side of the above equation: B + D – B = E + B – B which makes D = E
Hint #3
Add B to both sides of eq.5: A + B + B = C – B + B which becomes eq.5a) A + 2×B = C In eq.2, replace C with A + 2×B (from eq.5a): A + 2×B = A + F Subtract A from both sides of the above equation: A + 2×B – A = A + F – A which makes 2×B = F
Hint #4
Substitute 2×B for F in eq.3: E = A + B + 2×B which makes E = A + 3×B and also makes eq.3a) D = E = A + 3×B
Hint #5
Substitute A + 3×B for D and E (from eq.3a), and A + 2×B for C (from eq.5a) in eq.6: A + 3×B + A + 3×B – A = A – B + A + 2×B which becomes A + 6×B = 2×A + B Subtract A and B from both sides of the above equation: A + 6×B – A – B = 2×A + B – A – B which makes 5×B = A
Hint #6
Substitute 5×B for A in eq.3a: D = E = 5×B + 3×B which makes D = E = 8×B
Hint #7
Substitute 5×B for A in eq.5a: 5×B + 2×B = C which makes 7×B = C
Solution
Substitute 5×B for A, 7×B for C, 8×B for D and E, and 2×B for F in eq.1: 5×B + B + 7×B + 8×B + 8×B + 2×B = 31 which simplifies to 31×B = 31 Divide both sides of the above equation by 31: 31×B ÷ 31 = 31 ÷ 31 which means B = 1 making A = 5×B = 5 × 1 = 5 C = 7×B = 7 × 1 = 7 D = E = 8×B = 8 × 1 = 8 F = 2×B = 2 × 1 = 2 and ABCDEF = 517882