Puzzle for November 30, 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.6, replace F with D + E (from eq.3): B + E = A + D + E – C In the above equation, subtract E from both sides, and add C to both sides: B + E – E + C = A + D + E – C – E + C which becomes eq.6a) B + C = A + D
Hint #2
In eq.4, replace B + C with A + D (from eq.6a): D + F = A + D Subtract D from each side of the equation above: D + F – D = A + D – D which makes F = A
Hint #3
In eq.4, substitute A for F, and A + B for C (from eq.2): D + A = B + A + B which becomes D + A = 2×B + A Subtract A from both sides of the equation above: D + A – A = 2×B + A – A which makes D = 2×B
Hint #4
Substitute 2×B for D in eq.5: E – 2×B = B + 2×B which becomes E – 2×B = 3×B Add 2×B to both sides of the above equation: E – 2×B + 2×B = 3×B + 2×B which makes E = 5×B
Hint #5
Substitute 2×B for D, and 5×B for E in eq.3: F = 2×B + 5×B which makes F = 7×B and also makes A = F = 7×B
Hint #6
Substitute 7×B for A in eq.2: C = 7×B + B which makes C = 8×B
Solution
Substitute 7×B for A and F, 8×B for C, 2×B for D, and 5×B for E in eq.1: 7×B + B + 8×B + 2×B + 5×B + 7×B = 30 which simplifies to 30×B = 30 Divide both sides of the above equation by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = F = 7×B = 7×1 = 7 C = 8×B = 8×1 = 8 D = 2×B = 2×1 = 2 E = 5×B = 5×1 = 5 and ABCDEF = 718257