Puzzle for July 8, 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.
Scratchpad
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Hint #1
In eq.4, replace F with A + D (from eq.2): A + A + D = C + D Subtract D from both sides of the above equation: A + A + D - D = C + D - D which makes 2×A = C
Hint #2
In eq.5, replace D + E with C (from eq.3): C + F = A + C Subtract C from both sides of the above equation: C + F - C = A + C - C which makes F = A
Hint #3
In eq.2, substitute A for F: A = A + D Subtract A from both sides of the above equation: A - A = A + D - A which means 0 = D
Hint #4
Substitute 0 for D in eq.3: C = 0 + E which makes C = E and also makes E = C = 2×A
Hint #5
Substitute 0 for D, A for F, and 2×A for C and E in eq.6: B + 0 - A = A + 2×A + 2×A + A which becomes B - A = 6×A Add A to both sides: B - A + A = 6×A + A which makes B = 7×A
Solution
Substitute 7×A for B, 2×A for C and E, 0 for D, and A for F in eq.1: A + 7×A + 2×A + 0 + 2×A + A = 13 which simplifies to 13×A = 13 Divide both sides by 13: 13×A ÷ 13 = 13 ÷ 13 which means A = 1 making B = 7×A = 7 × 1 = 7 C = E = 2×A = 2 × 1 = 2 F = A = 1 and ABCDEF = 172021