Puzzle for January 7, 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
Help Area
Hint #1
In eq.5, replace C + D with A + E (from eq.3): A + B = A + E + E Subtract A from each side of the above equation: A + B - A = A + E + E - A which makes eq.5a) B = 2×E
Hint #2
In eq.4, substitute B + C for D (from eq.2): A + C = B + B + C Subtract C from each side of the above equation: A + C - C = B + B + C - C which simplifies to A = 2×B Substitute 2×E for B (from eq.5a): A = 2×2×E which makes A = 4×E
Hint #3
In eq.5, replace D with B + C (from eq.2): A + B = C + B + C + E Subtract B from both sides of the above equation: A + B - B = C + B + C + E - B which becomes eq.5b) A = 2×C + E
Hint #4
In eq.5b, replace A with 4×E: 4×E = 2×C + E Subtract E from both sides: 4×E - E = 2×C + E - E which becomes 3×E = 2×C Divide both sides by 2: 3×E ÷ 2 = 2×C ÷ 2 which makes 1½×E = C
Hint #5
Substitute 2×E for B, and 1½×E for C in eq.2: 2×E + 1½×E = D which makes 3½×E = D
Solution
Substitute 4×E for A, 2×E for B, 1½×E for C, and 3½×E for D in eq.1: 4×E + 2×E + 1½×E + 3½×E + E = 24 which simplifies to 12×E = 24 Divide both sides by 12: 12×E ÷ 12 = 24 ÷ 12 which makes E = 2 making A = 4×E = 4 × 2 = 8 B = 2×E = 2 × 2 = 4 C = 1½×E = 1½ × 2 = 3 D = 3½×E = 3½ × 2 = 7 and ABCDE = 84372