Puzzle for April 12, 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
Help Area
Hint #1
Add F to both sides of eq.5: B + F + F = A – F + F which becomes B + 2×F = A In eq.2, replace A with B + 2×F: B + C = B + 2×F + F which becomes B + C = B + 3×F Subtract B from each side of the above equation: B + C – B = B + 3×F – B which makes C = 3×F
Hint #2
In eq.6, replace C with 3×F: 3×F + F = B which makes 4×F = B
Hint #3
In eq.2, substitute 4×F for B, and 3×F for C: 4×F + 3×F = A + F which becomes 7×F = A + F Subtract F from each side of the equation above: 7×F – F = A + F – F which makes 6×F = A
Hint #4
Substitute 6×F for A, and 3×F for C in eq.4: 6×F + 3×F = D which makes 9×F = D
Hint #5
Substitute 9×F for D, 6×F for A, and 4×F for B in eq.3: 9×F + E = 6×F + 4×F which becomes 9×F + E = 10×F Subtract 9×F from each side of the above equation: 9×F + E – 9×F = 10×F – 9×F which makes E = F
Solution
Substitute 6×F for A, 4×F for B, 3×F for C, 9×F for D, and F for E in eq.1: 6×F + 4×F + 3×F + 9×F + F + F = 24 which simplifies to 24×F = 24 Divide both sides of the equation above by 24: 24×F ÷ 24 = 24 ÷ 24 which means F = 1 making A = 6×F = 6 × 1 = 6 B = 4×F = 4 × 1 = 4 C = 3×F = 3 × 1 = 3 D = 9×F = 9 × 1 = 9 E = F = 1 and ABCDEF = 643911