Puzzle for April 6, 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
Help Area
Hint #1
Subtract B, C, and E from both sides of eq.4: B + D - E - B - C - E = A + C + E - B - C - E which becomes D - C - 2×E = A - B In eq.3, replace A - B with D - C - 2×E: eq.3a) E - D = D - C - 2×E
Hint #2
Add (C + D - E) to each side of eq.3a: E - D + (C + D - E) = D - C - 2×E + (C + D - E) which simplifies to C = 2×D - 3×E In eq.2, replace C with 2×D - 3×E: D + E = 2×D - 3×E Add (3×E - D) to each side of the equation above: D + E + (3×E - D) = 2×D - 3×E + (3×E - D) which makes 4×E = D
Hint #3
In eq.2, substitute 4×E for D: 4×E + E = C which makes 5×E = C
Hint #4
eq.5 may be written as: 10×C + D = A + B + 10×D + E Subtract D from both sides of the above equation: 10×C + D - D = A + B + 10×D + E - D which becomes 10×C = A + B + 9×D + E Substitute (5×E) for C, and (4×E) for D: 10×(5×E) = A + B + 9×(4×E) + E which is equivalent to 50×E = A + B + 36×E + E which becomes 50×E = A + B + 37×E Subtract 37×E from each side: 50×E - 37×E = A + B + 37×E - 37×E which becomes eq.5a) 13×E = A + B
Hint #5
Add the left and right sides of eq.5a to the left and right sides of eq.3, respectively: E - D + 13×E = A - B + (A + B) which makes 14×E - D = 2×A Substitute 4×E for D: 14×E - 4×E = 2×A which becomes 10×E = 2×A Divide both sides by 2: 10×E ÷ 2 = 2×A ÷ 2 which makes 5×E = A
Hint #6
Substitute 5×E for A in eq.5a: 13×E = 5×E + B Subtract 5×E from each side: 13×E - 5×E = 5×E + B - 5×E which means 8×E = B
Solution
Substitute 5×E for A and C, 8×E for B, and 4×E for D in eq.1: 5×E + 8×E + 5×E + 4×E + E = 23 which simplifies to 23×E = 23 Divide both sides by 2: 23×E ÷ 23 = 23 ÷ 23 which makes E = 1 making A = C = 5×E = 5 × 1 = 5 B = 8×E = 8 × 1 = 8 D = 4×E = 4 × 1 = 4 and ABCDE = 58541