Puzzle for February 5, 2022 ( )
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.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
Multiply both sides of eq.5 by E: C × E = AB ÷ E × E which becomes eq.5a) C × E = AB In eq.4, substitute (C × E) for AB (from eq.5a): eq.4a) (C × E) + C = EF
Hint #2
In eq.6, replace EF with (C × E) + C (from eq.4a), and A + B with C (from eq.1): (C × E) + C = C + (C × D) Subtract C from each side of the above equation: (C × E) + C – C = C + (C × D) – C which becomes C × E = C × D Divide both sides by C: (C × E) ÷ C = (C × D) ÷ C which makes E = D
Hint #3
Substitute D for E in eq.3: D + F = A + D Subtract D from each side of the equation above: D + F – D = A + D – D which makes F = A
Hint #4
In eq.2, substitute E for D: F = C – E Add E to both sides of the above equation: F + E = C – E + E which becomes eq.2a) F + E = C
Hint #5
Substitute C × E for AB (from eq.5a), and F + E for C (from eq.2a) in eq.4: C × E + F + E = EF which may be written as C × E + F + E = 10×E + F Subtract E and F from both sides of the above equation: C × E + F + E – E – F = 10×E + F – E – F which becomes C × E = 9×E Divide both sides by E: C × E ÷ E = 9×E ÷ E which makes C = 9
Hint #6
Substitute 9 for C in eq.1: eq.1a) 9 = A + B
Hint #7
eq.4 may be written as: 10×A + B + C = 10×E + F which could be written as 9×A + A + B + C = 10×E + F Substitute 9 for A + B (from eq.1a), 9 for C, and A for F in the above equation: 9×A + 9 + 9 = 10×E + A which becomes 8×A + 18 = 10×E Divide both sides of the above equation by 2: (8×A + 18) ÷ 2 = 10×E ÷ 2 which becomes eq.4b) 4×A + 9 = 5×E
Hint #8
Substitute A for F, and 9 for C in eq.2a: eq.2b) A + E = 9
Hint #9
Substitute A + E for 9 (from eq.2b) into eq.4b: 4×A + A + E = 5×E which becomes 5×A + E = 5×E Subtract E from each side of the equation above: 5×A + E – E = 5×E – E which makes 5×A = 4×E Divide both sides of the above equation by 4: 5×A ÷ 4 = 4×E ÷ 4 which makes eq.4c) 1¼×A = E
Hint #10
Substitute 1¼×A for E in eq.2b: 1¼×A + A = 9 which makes 2¼×A = 9 Divide both sides of the above equation by 2¼: 2¼×A ÷ 2¼ = 9 ÷ 2¼ which makes A = 4 and also makes F = A = 4
Hint #11
Substitute 4 for A in eq.4c: 1¼ × 4 = E which makes 5 = E and also makes 5 = E = D
Solution
Substitute 4 for A in eq.1a: 9 = 4 + B Subtract 4 from both sides of the above equation: 9 – 4 = 4 + B – 4 which makes 5 = B and makes ABCDEF = 459554