Puzzle for April 24, 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.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
Add both E and F to each side of eq.6: B - E + E + F = C - F + E + F which becomes B + F = C + E In eq.5, replace C + E with B + F: B + F = A + D + F Subtract F from both sides of the above equation: B + F - F = A + D + F - F which becomes eq.6a) B = A + D
Hint #2
In eq.3, replace B with A + D (from eq.6a): D + F = A + D Subtract D from both sides of the equation above: D + F = A + D which makes F = A
Hint #3
In eq.3, substitute A for F: D + A = B Substitute D + E for A (from eq.2) in the above equation: D + D + E = B which becomes eq.3a) 2×D + E = B
Hint #4
eq.4 may be written as: 10×A + B - E = A + 10×E + F Substitute A for F in the above equation: 10×A + B - E = A + 10×E + A which is the same as 10×A + B - E = 2×A + 10×E Add (E - 2×A) to both sides: 10×A + B - E + (E - 2×A) = 2×A + 10×E + (E - 2×A) which becomes eq.4a) 8×A + B = 11×E
Hint #5
Substitute (D + E) for A (from eq.2), and 2×D + E for B (from eq.3a) in eq.4a: 8×(D + E) + 2×D + E = 11×E which may be written as 8×D + 8×E + 2×D + E = 11×E which becomes 10×D + 9×E = 11×E Subtract 9×E from both sides: 10×D + 9×E - 9×E = 11×E - 9×E which simplifies to 10×D = 2×E Divide both sides by 2: 10×D ÷ 2 = 2×E ÷ 2 which makes 5×D = E
Hint #6
Substitute 5×D for E in eq.3a: 2×D + 5×D = B which makes 7×D = B
Hint #7
Substitute 5×D for E in eq.2: A = D + 5×D which makes A = 6×D and which also makes F = A = 6×D
Hint #8
Substitute 5×D for E, and 6×D for A and F in eq.5: C + 5×D = 6×D + D + 6×D which becomes C + 5×D = 13×D Subtract 5×D from both sides: C + 5×D - 5×D = 13×D - 5×D which makes C = 8×D
Solution
Substitute 6×D for A and F, 7×D for B, 8×D for C, and 5×D for E in eq.1: 6×D + 7×D + 8×D + D + 5×D + 6×D = 33 which becomes 33×D = 33 Divide both sides by 33: 33×D ÷ 33 = 33 ÷ 33 which means D = 1 making A = F = 6×D = 6 × 1 = 6 B = 7×D = 7 × 1 = 7 C = 8×D = 8 × 1 = 8 E = 5×D = 5 × 1 = 5 and ABCDEF = 678156