Puzzle for May 30, 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 non-negative integer.
Scratchpad
Help Area
Hint #1
In eq.6, replace B + D with E + F (from eq.5): C + E = E + F + F which becomes C + E = E + 2×F Subtract E from each side of the equation above: C + E – E = E + 2×F – E which makes C = 2×F
Hint #2
In eq.2, replace C with 2×F: D = 2×F + F which makes D = 3×F
Hint #3
In eq.3, substitute 2×F for C, and 3×F for D: B = 2×F + 3×F which makes B = 5×F
Hint #4
Substitute 5×F for B, and 2×F for C in eq.4: A = 5×F + 2×F which makes A = 7×F
Hint #5
Substitute 5×F for B, and 3×F for D in eq.5: E + F = 5×F + 3×F which becomes E + F = 8×F Subtract F from each side of the equation above: E + F – F = 8×F – F which makes E = 7×F
Solution
Substitute 7×F for A and E, 5×F for B, 2×F for C, and 3×F for D in eq.1: 7×F + 5×F + 2×F + 3×F + 7×F + F = 25 which simplifies to 25×F = 25 Divide both sides of the above equation by 25: 25×F ÷ 25 = 25 ÷ 25 which means F = 1 making A = 7×F = 7 × 1 = 7 B = 5×F = 5 × 1 = 5 C = 2×F = 2 × 1 = 2 D = 3×F = 3 × 1 = 3 E = 7×F = 7 × 1 = 7 and ABCDEF = 752371