Puzzle for April 5, 2021 ( )
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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 positive integer.
Our thanks go out to Judah S (age 14) for this puzzle! Thank you, Judah!
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Hint #1
In eq.5, replace B with C + E (from eq.2): E – C = C + E – E which becomes E – C = C Add C to both sides of the above equation: E – C + C = C + C which makes E = 2×C
Hint #2
In eq.2, replace E with 2×C: C + 2×C = B which makes 3×C = B
Hint #3
In eq.3, substitute 3×C for B: F = 3×C + C which makes F = 4×C
Hint #4
Substitute 3×C for B, and 4×C for F in eq.4: D = 3×C + 4×C which makes D = 7×C
Hint #5
Substitute 7×C for D in eq.1: A + C = 7×C Subtract C from each side of the above equation: A + C – C = 7×C – C which makes A = 6×C
Solution
Substitute 3×C for B, 2×C for E, and 6×C for A in eq.6: 3×C × 2×C = 6×C which becomes 6×C² = 6×C Divide both sides of the above equation by 6×C: 6×C² ÷ 6×C = 6×C ÷ 6×C which means C = 1 making A = 6×C = 6 × 1 = 6 B = 3×C = 3 × 1 = 3 D = 7×C = 7 × 1 = 7 E = 2×C = 2 × 1 = 2 F = 4×C = 4 × 1 = 4 and ABCDEF = 631724