Puzzle for May 22, 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 positive integer.
Scratchpad
Help Area
Hint #1
Add A to both sides of eq.3: A + E + A = B + C - A + A which becomes eq.3a) 2×A + E = B + C In eq.6, replace B + C with 2×A + E (from eq.3a): 2×A + E = D + E + F Subtract E from both sides of the above equation: 2×A + E - E = D + E + F - E which becomes eq.6a) 2×A = D + F
Hint #2
In eq.4, replace D + F with 2×A (from eq.6a): 2×A = A + E Subtract A from each side of the above equation: 2×A - A = A + E - A which makes A = E
Hint #3
In eq.5, replace E with A: A × F = A Divide both sides of the above equation by A: A × F ÷ A = A ÷ A which makes F = 1
Hint #4
In eq.6a, substitute 1 for F: 2×A = D + 1 Subtract 1 from each side of the above equation: 2×A - 1 = D + 1 - 1 which becomes eq.6b) 2×A - 1 = D
Hint #5
Substitute 1 for F, and 2×A - 1 for D (from eq.6b) in eq.2: B + 1 = 2×A - 1 Subtract 1 from each side: B + 1 - 1 = 2×A - 1 - 1 which makes eq.2a) B = 2×A - 2
Hint #6
Substitute A for E, and 2×A - 2 for B (from eq.2a) in eq.3a: 2×A + A = 2×A - 2 + C In the equation above, subtract 2×A from each side, and add 2 to each side: 2×A + A - 2×A + 2 = 2×A - 2 + C - 2×A + 2 which makes eq.3b) A + 2 = C
Solution
Substitute 2×A - 2 for B (from eq.2a), A + 2 for C (from eq.3b), 2×A - 1 for D (from eq.6b), A for E, and 1 for F in eq.1: A + 2×A - 2 + A + 2 + 2×A - 1 + A + 1 = 30 which simplifies to 7×A = 35 Divide both sides of the above equation by 7: 7×A ÷ 7 = 35 ÷ 7 which means A = 5 making B = 2×A - 2 = 2×5 - 2 = 10 - 2 = 8 (from eq.2a) C = A + 2 = 5 + 2 = 7 (from eq.3b) D = 2×A - 1 = 2×5 - 1 = 9 (from eq.6b) A = E = 5 and ABCDDEF = 587951