Puzzle for June 3, 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.
Scratchpad
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Hint #1
eq.3 may be written as: F = (B + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (B + D + E) ÷ 3 which becomes eq.3a) 3×F = B + D + E
Hint #2
In eq.5, replace B + D + E with 3×F (from eq.3a): E × F = 3×F Divide both sides of the equation above by F: E × F ÷ F = 3×F ÷ F which makes E = 3
Hint #3
In eq.2, substitute 3 for E: 3 + F = B + D – F Add F to both sides of the the above equation: 3 + F + F = B + D – F + F which becomes eq.2a) 3 + 2×F = B + D
Hint #4
In eq.3a, substitute 3 + 2×F for B + D (from eq.2a), and 3 for E: 3×F = 3 + 2×F + 3 which becomes 3×F = 2×F + 6 Subtract 2×F from both sides of the equation above: 3×F – 2×F = 2×F + 6 – 2×F which makes F = 6
Hint #5
Substitute 6 for F in eq.2a: 3 + 2×6 = B + D which becomes 3 + 12 = B + D which becomes eq.2b) 15 = B + D
Hint #6
In eq.4, substitute 3 for E, 15 for B + D (from eq.2b), and 6 for F: B × 3 = 15 + 6 which becomes 3×B = 21 Divide both sides of the above equation by 3: 3×B ÷ 3 = 21 ÷ 3 which makes B = 7
Hint #7
Substitute 7 for B in eq.2b: 15 = 7 + D Subtract 7 from both sides of the equation above: 15 – 7 = 7 + D – 7 which makes 8 = D
Hint #8
Substitute 8 for D, and 3 for E in eq.1: A + 8 = C + 3 – A In the above equation, subtract 3 from both sides, and add A to both sides: A + 8 – 3 + A = C + 3 – A – 3 + A which becomes eq.1a) 2×A + 5 = C
Hint #9
In eq.6, substitute (2×A + 5) for C (from eq.1a), and 7 for B: (2×A + 5) ÷ 7 = A Multiply both sides of the above equation by 7: (2×A + 5) ÷ 7 × 7 = A × 7 which becomes 2×A + 5 = 7×A Subtract 2×A from both sides: 2×A + 5 – 2×A = 7×A – 2×A which makes 5 = 5×A Divide both sides by 5: 5 ÷ 5 = 5×A ÷ 5 which means 1 = A
Solution
Substitute 1 for A in eq.1a: 2×1 + 5 = C which becomes 2 + 5 = C which makes 7 = C and makes ABCDEF = 177836