Puzzle for January 14, 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.
* EF and AB are 2-digit numbers (not E×F or A×B).
Scratchpad
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Hint #1
In eq.2, replace E with A + B (from eq.1): F = B + A + B which becomes eq.2a) F = A + 2×B
Hint #2
In eq.3, replace E with A + B (from eq.1), and replace F with A + 2×B (from eq.2a): B + D = A + A + B + A + 2×B which becomes B + D = 3×A + 3×B Subtract B from each side of the equation above: B + D – B = 3×A + 3×B – B which becomes eq.3a) D = 3×A + 2×B
Hint #3
eq.4 may be written as: A + 10×E + F = 10×A + B + D – E In the equation above, subtract A from both sides, and add E to both sides: A + 10×E + F – A + E = 10×A + B + D – E – A + E which becomes eq.4a) 11×E + F = 9×A + B + D
Hint #4
In eq.4a, substitute (A + B) for E (from eq.1), A + 2×B for F (from eq.2a), and 3×A + 2×B for D (from eq.3a): 11×(A + B) + A + 2×B = 9×A + B + 3×A + 2×B which becomes 11×A + 11×B + A + 2×B = 12×A + 3×B which becomes 12×A + 13×B = 12×A + 3×B Subtract 12×A and 3×B from both sides of the above equation: 12×A + 13×B – 12×A – 3×B = 12×A + 3×B – 12×A – 3×B which simplifies to 10×B = 0 which means B = 0
Hint #5
Substitute 0 for B in eq.1: E = A + 0 which makes E = A
Hint #6
Substitute 0 for B in eq.2a: F = A + 2×0 which makes F = A
Hint #7
Substitute 0 for B in eq.3a: D = 3×A + 2×0 which makes D = 3×A
Hint #8
eq.5 may be written as: C × F = 10×A + B – D Substitute A for F, 0 for B, and 3×A for D in the above equation: C × A = 10×A + 0 – 3×A which becomes C × A = 7×A Since A ≠ 0 (from eq.6), divide both sides by A: C × A ÷ A = 7×A ÷ A which makes C = 7
Solution
Substitute 3×A for D, 7 for C, and A for E and F in eq.6: 3×A – 7 = (A ÷ A) + (A ÷ A) which becomes 3×A – 7 = 1 + 1 which becomes 3×A – 7 = 2 Add 7 to both sides of the above equation: 3×A – 7 + 7 = 2 + 7 which makes 3×A = 9 Divide both sides by 3: 3×A ÷ 3 = 9 ÷ 3 which means A = 3 making D = 3×A = 3×3 = 9 E = F = A = 3 and ABCDEF = 307933