Puzzle for May 19, 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
Subtract F from both sides of eq.2: D – A – F = E + F – F which becomes D – A – F = E which may be written as D – F – A = E In the above equation, replace D – F with A + C (from eq.6): A + C – A = E which becomes C = E
Hint #2
In eq.4, replace E with C: C – F = C – B Subtract C from each side of the equation above: C – F – C = C – B – C which makes –F = –B which means F = B
Hint #3
In eq.5, substitute C for E, and B for F: C + C = A – B which becomes 2×C = A – B Add B to both sides of the above equation: 2×C + B = A – B + B which becomes eq.5a) 2×C + B = A
Hint #4
In eq.3, substitute B for F, and 2×C + B for A (from eq.5a): B + B = 2×C + B + C which becomes 2×B = 3×C + B Subtract B from both sides of the above equation: 2×B – B = 3×C + B – B which makes B = 3×C and also makes F = B = 3×C
Hint #5
Substitute 3×C for B in eq.5a: 2×C + 3×C = A which makes 5×C = A
Hint #6
Substitute 5×C for A, C for E, and 3×C for F in eq.2: D – 5×C = C + 3×C which becomes D – 5×C = 4×C Add 5×C to both sides of the above equation: D – 5×C + 5×C = 4×C + 5×C which makes D = 9×C
Solution
Substitute 5×C for A, 3×C for B and F, 9×C for D, and C for E in eq.1: 5×C + 3×C + C + 9×C + C + 3×C = 22 which simplifies to 22×C = 22 Divide both sides of the above equation by 22: 22×C ÷ 22 = 22 ÷ 22 which means C = 1 making A = 5×C = 5 × 1 = 5 B = F = 3×C = 3 × 1 = 3 D = 9×C = 9 × 1 = 9 E = C = 1 and ABCDEF = 531913