Puzzle for May 28, 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.
* BC and DE are 2-digit numbers (not B×C or D×E).
Scratchpad
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Hint #1
eq.6 may be written as: 10×D + E = A + D + E Subtract D and E from both sides of the above equation: 10×D + E – D – E = A + D + E – D – E which makes 9×D = A
Hint #2
Add A and E to both sides of eq.3: D – A + A + E = B – E + A + E which becomes eq.3a) D + E = B + A In eq.3a above, replace A with C + E (from eq.2): D + E = B + C + E Subtract E from each side of the above equation: D + E – E = B + C + E – E which becomes eq.3b) D = B + C
Hint #3
eq.4 may be written as: B + C + D + F = A – (B + C) – F In the above equation, replace B + C with D (from eq.3b), and A with 9×D: D + D + F = 9×D – (D) – F which becomes 2×D + F = 8×D – F Subtract 2×D from both sides, and add F to both sides: 2×D + F – 2×D + F = 8×D – F – 2×D + F which becomes 2×F = 6×D Divide both sides by 2: 2×F ÷ 2 = 6×D ÷ 2 which makes F = 3×D
Hint #4
eq.5 may be written as: 10×B + C = D + E which may be written as 9×B + B + C = D + E In the above equation, substitute D for B + C (from eq.3b): 9×B + D = D + E Subtract D from each side: 9×B + D – D = D + E – D which makes eq.5a) 9×B = E
Hint #5
Substitute 9×B for E, and 9×D for A in eq.3a: D + 9×B = B + 9×D Subtract both B and D from each side of the above equation: D + 9×B – B – D = B + 9×D – B – D which becomes 8×B = 8×D Divide each side by 8: 8×B ÷ 8 = 8×D ÷ 8 which makes B = D which also makes
Hint #6
Substitute D for B in 5a: 9×D = E
Hint #7
Substitute B for D in eq.3b: B = B + C Subtract B from both sides of the equation above: B – B = B + C – B which means 0 = C
Solution
Substitute 9×D for A and E, D for B, 0 for C, and 3×D for F in eq.1: 9×D + D + 0 + D + 9×D + 3×D = 23 which simplifies to 23×D = 23 Divide both sides of the equation above by 23: 23×D ÷ 23 = 23 ÷ 23 which means D = 1 making A = E = 9×D = 9 × 1 = 9 B = D = 1 F = 3×D = 3 × 1 = 3 and ABCDEF = 910193