Puzzle for September 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.
* AB, BC, and DE are 2-digit numbers (not A×B, B×C, or D×E).
Scratchpad
Help Area
Hint #1
In eq.3, add D to both sides, and subtract A from each side: C – D + D – A = A + D + D – A which becomes eq.3a) C – A = 2×D In eq.4, add D to both sides, and subtract C from each side: B – D + E + D – C = C + D + D – C which becomes eq.4a) B + E – C = 2×D
Hint #2
In eq.4a, replace 2×D with C – A (from eq.3a): B + E – C = C – A In the equation above, add A and C to both sides, and subtract B from both sides: B + E – C + A + C – B = C – A + A + C – B which simplifies to E + A = 2×C – B which may be written as eq.4b) A + E = 2×C – B
Hint #3
In eq.4b, replace A + E with B (from eq.2): B = 2×C – B Add B to both sides of the above equation: B + B = 2×C – B + B which makes 2×B = 2×C Divide both sides by 2: 2×B ÷ 2 = 2×C ÷ 2 which means B = C
Hint #4
In eq.4, substitute C for B: C – D + E = C + D In the equation above, add D to both sides, and subtract C from both sides: C – D + E + D – C = C + D + D – C which makes E = 2×D
Hint #5
eq.5 may be written as: 10×D + E + B = 10×B + C – (10×A + B) which is the same as 10×D + E + B = 10×B + C – 10×A – B which becomes 10×D + E + B = 9×B + C – 10×A Subtract B from both sides: 10×D + E + B – B = 9×B + C – 10×A – B which becomes 10×D + E = 8×B + C – 10×A which may be written as 5×(2×D) + E = 8×B + C – 10×A Substitute E for 2×D, and B for C in the above equation: 5×(E) + E = 8×B + B – 10×A which becomes eq.5a) 6×E = 9×B – 10×A
Hint #6
Substitute A + E for B (from eq.2) in eq.5a: 6×E = 9×(A + E) – 10×A which is equivalent to 6×E = 9×A + 9×E – 10×A which becomes 6×E = 9×E – A Add A to both sides, and subtract 6×E from both sides: 6×E + A – 6×E = 9×E – A + A – 6×E which makes A = 3×E Substitute (2×D) for E: A = 3×(2×D) which means A = 6×D
Hint #7
Substitute 6×D for A, and 2×D for E in eq.2: 6×D + 2×D = B which makes 8×D = B and also makes C = B = 8×D
Solution
Substitute 6×D for A, 8×D for B and C, and 2×D for E in eq.1: 6×D + 8×D + 8×D + D + 2×D = 25 which simplifies to 25×D = 25 Divide both sides by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = 6×D = 6 × 1 = 6 B = C = 8×D = 8 × D = 8 E = 2×D = 2 × 1 = 2 and ABCDE = 68812