Puzzle for June 19, 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 is a 2-digit number (not A×B).
Scratchpad
Help Area
Hint #1
Subtract the left and right sides of eq.3 from the left and right sides of eq.2, respectively: B + D - (D + E) = A + C - C which is equivalent to B + D - D - E = A which becomes B - E = A Add E to both sides of the above equation: B - E + E = A + E which becomes eq.2a) B = A + E
Hint #2
In eq.4, replace B with A + E (from eq.2a): F = A + E + E which becomes eq.4a) F = A + 2×E
Hint #3
In eq.5, replace F with A + 2×E (from eq.4a): E + A + 2×E = D which becomes eq.5a) A + 3×E = D
Hint #4
In eq.3, replace D with A + 3×E (from eq.5a): A + 3×E + E = C which becomes eq.3a) A + 4×E = C
Hint #5
eq.6 may be written as: C + E = 10×A + B - E Substitute A + 4×E for C (from eq.3a), and A + E for B (from eq.2a) in the above equation: A + 4×E + E = 10×A + A + E - E which becomes A + 5×E = 11A Subtract A from both sides: A + 5×E - A = 11×A - A which means 5×E = 10×A Divide both sides by 5: 5×E ÷ 5 = 10×A ÷ 5 which makes E = 2×A
Hint #6
Substitute 2×A for E in eq.2a: B = A + 2×A which makes B = 3×A
Hint #7
Substitute (2×A) for E in eq.3a: A + 4×(2×A) = C which is equivalent to A + 8×A = C which makes 9×A = C
Hint #8
Substitute (2×A) for E in eq.5a: A + 3×(2×A) = D which becomes A + 6×A = D which makes 7×A = D
Hint #9
Substitute (2×A) for E in eq.4a: F = A + 2×(2×A) which becomes F = A + 4×A which makes F = 5×A
Solution
Substitute 3×A for B, 9×A for C, 7×A for D, 2×A for E, and 5×A for F in eq.1: A + 3×A + 9×A + 7×A + 2×A + 5×A = 27 which simplifies to 27×A = 27 Divide both sides by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 3×A = 3 × 1 = 3 C = 9×A = 9 × 1 = 9 D = 7×A = 7 × 1 = 7 E = 2×A = 2 × 1 = 2 F = 5×A = 5 × 1 = 5 and ABCDEF = 139725