Puzzle for July 25, 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.
* CD and DE are 2-digit numbers (not C×D or D×E).
Scratchpad
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Hint #1
Add D to both sides of eq.4: B - D + D = C + E + D which becomes B = C + E + D which may be written as B = C + D + E In eq.2, replace C + D + E with B: A = B
Hint #2
Add C and E to both sides of eq.3: D - E + C + E = E - C + C + E which becomes D + C = 2×E which may be written as eq.3a) C + D = 2×E In eq.2, replace C + D with 2×E: A = 2×E + E which makes A = 3×E which also makes B = A = 3×E
Hint #3
eq.5 may be written as: 10×D + E - (10×C + D) = A + B - C + E which is equivalent to 10×D + E - 10×C - D = A + B - C + E In the above equation, subtract E from each side, and add C to each side: 10×D + E - 10×C - D - E + C = A + B - C + E - E + C which simplifies to 9×D - 9×C = A + B Substitute 3×E for both A and B: 9×D - 9×C = 3×E + 3×E which becomes 9×D - 9×C = 6×E which may be written as eq.5a) 9×D - 9×C = 3×(2×E)
Hint #4
Substitute C + D for 2×E (from eq.3a) in eq.5a: 9×D - 9×C = 3×(C + D) which is the same as 9×D - 9×C = 3×C + 3×D In the above equation, add 9×C to both sides, and subtract 3×D from both sides: 9×D - 9×C + 9×C - 3×D = 3×C + 3×D + 9×C - 3×D which simplifies to 6×D = 12×C Divide both sides by 6: 6×D ÷ 6 = 12×C ÷ 6 which means D = 2×C
Hint #5
Substitute 2×C for D in eq.3a: C + 2×C = 2×E which becomes 3×C = 2×E Divide both sides by 2: 3×C ÷ 2 = 2×E ÷ 2 which becomes 1½×C = E
Hint #6
Substitute 2×C for D, and 1½×C for E in eq.2: A = C + 2×C + 1½×C which makes A = 4½×C and which also makes B = A = 4½×C
Solution
Substitute 4½×C for A and B, 2×C for D, and 1½×C for E in eq.1: 4½×C + 4½×C + C + 2×C + 1½×C = 27 which simplifies to 13½×C = 27 Divide both sides of the above equation by 13½: 13½×C ÷ 13½ = 27 ÷ 13½ which means C = 2 making A = B = 4½×C = 4½ × 2 = 9 D = 2×C = 2 × 2 = 4 E = 1½×C = 1½ × 2 = 3 and ABCDE = 99243