Puzzle for April 25, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* BC and EF are 2-digit numbers (not B×C or E×F).
Scratchpad
Help Area
Hint #1
In eq.3, replace B with C + E (from eq.2): E + F = A + C + E Subtract E from both sides of the above equation: E + F - E = A + C + E - E which becomes eq.3a) F = A + C
Hint #2
eq.5 may be written as: 10×E + F = A + 10×B + C which may also be written as 10×E + F = A + C + 10×B In the above equation, replace A + C with F (from eq.3a): 10×E + F = F + 10×B Subtract F from both sides of the above equation: 10×E + F - F = F + 10×B - F which becomes 10×E = 10×B Divide both sides by 10: 10×E ÷ 10 = 10×B ÷ 10 which makes E = B
Hint #3
In eq.2, substitute B for E: C + B = B Subtract B from both sides of the above equation: C + B - B = B - B which makes C = 0
Hint #4
In eq.3a, substitute 0 for C: F = A + 0 which makes F = A
Hint #5
In eq.4, substitute A for F: A + A - D = B + D Subtract D from both sides of the equation above: A + A - D - D = B + D - D which becomes eq.4a) 2×A - 2×D = B
Hint #6
Substitute B for E, and A for F in eq.6: B + B - A = A + D + A which becomes 2×B - A = 2×A + D Substitute (2×A - 2×D) for B (from eq.4a) in the above equation: 2×(2×A - 2×D) - A = 2×A + D which is equivalent to 4×A - 4×D - A = 2×A + D which becomes 3×A - 4×D = 2×A + D Add 4×D to each side, and subtract 2×A from each side of the above equation: 3×A - 4×D + 4×D - 2×A = 2×A + D + 4×D - 2×A which makes A = 5×D and also makes F = A = 5×D
Hint #7
Substitute 5×D for A in eq.4a: 2×5×D - 2×D = B which becomes 10×D - 2×D = B which makes 8×D = B and also makes E = B = 8×D
Solution
Substitute 5×D for A and F, 8×D for B and E, and 0 for C in eq.1: 5×D + 8×D + 0 + D + 8×D + 5×D = 27 which simplifies to 27×D = 27 Divide both sides by 27: 27×D ÷ 27 = 27 ÷ 27 which makes D = 1 making A = F = 5×D = 5 × 1 = 5 B = E = 8×D = 8 × 1 = 8 and ABCDEF = 580185