Puzzle for May 10, 2022 ( )
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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.
Today, we are featuring another puzzle contributed by Judah S (age 15). Thank you so much for today's puzzle, Judah, and for every other puzzle this week, too!
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Hint #1
Add C to both sides of eq.3: B + E + C = F – C + C which becomes B + E + C = F which may be written as B + C + E = F In the above equation, replace B + C with E (from eq.1): E + E = F which makes 2×E = F
Hint #2
In eq.5, replace F with 2×E: 2×E = (E × E) – 2×E which may be written as 2×E = E² – 2×E Since E ≠ 0 (from eq.4), divide both sides of the above equation by E: (2×E) ÷ E = (E² – 2×E) ÷ E which becomes 2 = E – 2 Add 2 to both sides of the above equation: 2 + 2 = E – 2 + 2 which makes 4 = E and also makes F = 2×E = 2×4 = 8
Hint #3
In eq.4, substitute 8 for F, and 4 for E: C = 8 ÷ 4 which makes C = 2
Hint #4
Substitute 4 for E, and 2 for C in eq.1: 4 = B + 2 Subtract 2 from each side of the equation above: 4 – 2 = B + 2 – 2 which makes 2 = B
Hint #5
Substitute 2 for B and C in eq.2: D = 2 – 2 which makes D = 0
Solution
Substitute 0 for D, 4 for E, and 2 for B and C in eq.6: 0 + 4 = (A × 2) + 2 which becomes 4 = 2×A + 2 Subtract 2 from each side of the above equation: 4 – 2 = 2×A + 2 – 2 which makes 2 = 2×A Divide both sides by 2: 2 ÷ 2 = 2×A ÷ 2 which makes 1 = A and makes ABCDEF = 122048