Puzzle for March 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.2, replace A with C + F (from eq.3), and B with D + F (from eq.4): C + F + D + F = C + D + E which becomes C + D + 2×F = C + D + E Subtract C and D from each side of the above equation: C + D + 2×F – C – D = C + D + E – C – D which simplifies to 2×F = E
Hint #2
In eq.6, replace E with 2×F: D = 2×F + F which makes D = 3×F
Hint #3
In eq.4, replace D with 3×F: B = 3×F + F which makes B = 4×F
Hint #4
In eq.5, substitute 4×F for B, and 3×F for D: C = 4×F + 3×F which makes C = 7×F
Hint #5
Substitute 7×F for C in eq.3: A = 7×F + F which makes A = 8×F
Solution
Substitute 8×F for A, 4×F for B, 7×F for C, 3×F for D, and 2×F for E in eq.1: 8×F + 4×F + 7×F + 3×F + 2×F + F = 25 which simplifies to 25×F = 25 Divide both sides of the equation above by 25: 25×F ÷ 25 = 25 ÷ 25 which means F = 1 making A = 8×F = 8 × 1 = 8 B = 4×F = 4 × 1 = 4 C = 7×F = 7 × 1 = 7 D = 3×F = 3 × 1 = 3 E = 2×F = 2 × 1 = 2 and ABCDEF = 847321