Puzzle for June 14, 2023 ( )
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.
Once again, we thank Abby S (age 12) for sending us a fun and interesting puzzle! Thank you, Abby!
Scratchpad
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Hint #1
In eq.1, substitute (E × F) for B (from eq.4): E = (E × F) - F Add F to both sides of the above equation: E + F = (E × F) - F + F which becomes eq.1a) E + F = E × F
Hint #2
In eq.1a, substitute (D × E) for F (from eq.3): E + (D × E) = E × (D × E) Divide both sides of the above equation by E: (E + (D × E)) ÷ E = (E × (D × E)) ÷ E which becomes 1 + D = D × E Divide both sides by D: (1 + D) ÷ D = (D × E) ÷ D which becomes eq.1b) (1 ÷ D) + 1 = E
Hint #3
To make eq.1b true, check several possible values for D and E: If D = 1, then E = (1 ÷ 1) + 1 = 1 + 1 = 2 If D = 2, then E = (1 ÷ 2) + 1 = ½ + 1 = 1½ If D = 3, then E = (1 ÷ 3) + 1 = ⅓ + 1 = 1⅓ If D = 4, then E = (1 ÷ 4) + 1 = ¼ + 1 = 1¼ If D > 4, then 1 < E < 2 Since E must be a positive integer, then the above equations make: E = 2 which makes D = 1
Hint #4
In eq.1a, replace E with 2: 2 + F = 2 × F Subtract F from each side of the equation above: 2 + F - F = (2 × F) - F which makes 2 = F
Hint #5
In eq.4, replace E with 2, and F with 2: B = 2 × 2 which makes B = 4
Hint #6
In eq.2, substitute 4 for B: eq.2a) C = A + 4
Hint #7
Substitute A + 4 for C (from eq.2a), 4 for B, 1 for D, and 2 for E and F in eq.5: A + A + 4 = (4 + 1 + 2) × 2 which becomes 2×A + 4 = 7 × 2 which becomes 2×A + 4 = 14 Subtract 4 from each side of the above equation: 2×A + 4 - 4 = 14 - 4 which makes 2×A = 10 Divide both sides by 2: 2×A ÷ 2 = 10 ÷ 2 which makes A = 5
Solution
Substitute 5 for A in eq.2a: C = 5 + 4 which makes C = 9 and makes ABCDEF = 549122