Puzzle for February 15, 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 non-negative integer.
Scratchpad
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Hint #1
In eq.6, replace D with C + F (from eq.2): B + C – F = C + F + F which becomes B + C – F = C + 2×F In the above equation, subtract C from both sides, and add F to both sides: B + C – F – C + F = C + 2×F – C + F which simplifies to B = 3×F
Hint #2
In eq.5, replace B with 3×F: E = 3×F + F which makes E = 4×F
Hint #3
In eq.4, substitute 3×F for B, and 4×F for E: C – F = 3×F + 4×F which becomes C – F = 7×F Add F to both sides of the above equation: C – F + F = 7×F + F which makes C = 8×F
Hint #4
Substitute 8×F for C in eq.2: D = 8×F + F which makes D = 9×F
Hint #5
Substitute 4×F for E in eq.3: A – 4×F = 4×F + F which becomes A – 4×F = 5×F Add 4×F to both sides of the equation above: A – 4×F + 4×F = 5×F + 4×F which makes A = 9×F
Solution
Substitute 9×F for A and D, 3×F for B, 8×F for C, and 4×F for E in eq.1: 9×F + 3×F + 8×F + 9×F + 4×F + F = 34 which simplifies to 34×F = 34 Divide both sides of the equation above by 34: 34×F ÷ 34 = 34 ÷ 34 which makes F = 1 and makes A = D = 9×F = 9 × 1 = 9 B = 3×F = 3 × 1 = 3 C = 8×F = 8 × 1 = 8 E = 4×F = 4 × 1 = 4 and ABCDEF = 938941