Puzzle for May 28, 2021 ( )
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.
Our thanks go out to Lily S (age 12) for sending us this puzzle. Thank you, Lily!
Scratchpad
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Hint #1
Divide each side of eq.3 by both D and C: ((D × E) ÷ D) ÷ C = ((A × C) ÷ D) ÷ C which simplifies to eq.3a) E ÷ C = A ÷ D Divide each side of eq.5 by both D and E: ((C × D) ÷ D) ÷ E = ((A × E) ÷ D) ÷ E which simplifies to eq.5a) C ÷ E = A ÷ D
Hint #2
In eq.5a, replace A ÷ D with E ÷ C (from eq.3a): C ÷ E = E ÷ C Multiply both sides of the above equation by E and C: ((C ÷ E) × E) × C) = ((E ÷ C) × E) × C) which simplifies to C² = E² which means C = E
Hint #3
In eq.5, replace E with C: C × D = A × C Divide each side of the above equation by C: (C × D) ÷ C = (A × C) ÷ C which makes D = A
Hint #4
In eq.4, substitute A for D: A × F = B × A Divide both sides of the above equation by A: (A × F) ÷ A = (B × A) ÷ A which makes F = B
Hint #5
Add C and B to both sides of eq.2: B – C + C + B = A – B + C + B which becomes eq.2a) 2×B = A + C Substitute D for A, and E for C in eq.2a: eq.2b) 2×B = D + E
Hint #6
eq.1 may be written as: B + F + A + C + D + E = 48 Substitute B for F, and 2×B for A + C (from eq.2a) and for D + E (from eq.2b) in the equation above: B + B + 2×B + 2×B = 48 which makes 6×B = 48 Divide both sides of the above equation by 6: 6×B ÷ 6 = 48 ÷ 6 which makes B = 8 and also makes F = B = 8
Hint #7
Substitute 8 for B and F, C for E, and A for D in eq.6: 8 – C + 8 – (A ÷ A) = C + (8 ÷ 8) which becomes 16 – C – (1) = C + (1) which becomes 15 – C = C + 1 In the equation above, add C to both sides, and subtract 1 from both sides: 15 – C + C – 1 = C + 1 + C – 1 which becomes 14 = 2×C Divide both sides by 2: 14 ÷ 2 = 2×C ÷ 2 which makes 7 = C and also makes E = C = 7
Solution
Substitute 8 for B, and 7 for C in eq.2a: 2×8 = A + 7 which becomes 16 = A + 7 Subtract 7 from both sides of the above equation: 16 – 7 = A + 7 – 7 which makes 9 = A and also makes D = A = 9 and makes ABCDEF = 987978