Puzzle for August 13, 2019 ( )
Scratchpad
Find the 5-digit number ABCDE by solving the following equations:
A, B, C, D, and E each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Add E to both sides of eq.3: C + D + E = A – E + E which becomes eq.3a) C + D + E = A In eq.2, replace A with C + D + E (from eq.3a): B + C + E = C + D + E + D Subtract C and E from both sides: B + C + E – C – E = C + D + E + D – C – E which simplifies to B = 2×D
Hint #2
In eq.5, replace A with C + D + E (from eq.3a), and replace B with 2×D: D + E = C + D + E – 2×D Subtract E from both sides of the above equation: D + E – E = C + D + E – 2×D – E which becomes D = C – D Add D to both sides: D + D = C – D + D which makes 2×D = C
Hint #3
In eq.4, substitute 2×D for both B and C: E – D = 2×D + 2×D + D which becomes E – D = 5×D Add D to both sides: E – D + D = 5×D + D which makes E = 6×D
Hint #4
Substitute 2×D for C, and 6×D for E in eq.3a: 2×D + D + 6×D = A which means 9×D = A
Solution
Substitute 9×D for A, 2×D for B and C, and 6×D for E in eq.1: 9×D + 2×D + 2×D + D + 6×D = 20 which simplifies to 20×D = 20 Divide both sides of the equation above by 20: 20×D ÷ 20 = 20 ÷ 20 which means D = 1 making A = 9×D = 9 × 1 = 9 B = C = 2×D = 2 × 1 = 2 E = 6×D = 6 × 1 = 6 and ABCDE = 92216