Puzzle for June 13, 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 D with C + F (from eq.5): C + E = C + F + F which becomes C + E = C + 2×F Subtract C from each side of the equation above: C + E – C = C + 2×F – C which makes E = 2×F
Hint #2
In eq.4, replace E with 2×F: C = 2×F + F which makes C = 3×F
Hint #3
In eq.5, substitute 3×F for C: D = 3×F + F which makes D = 4×F
Hint #4
In eq.2, substitute 3×F for C, and 4×F for D: A = 3×F + 4×F which makes A = 7×F
Hint #5
Substitute 7×F for A in eq.3: B = 7×F + F which makes B = 8×F
Solution
Substitute 7×F for A, 8×F for B, 3×F for C, 4×F for D, and 2×F for E in eq.1: 7×F + 8×F + 3×F + 4×F + 2×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 = 8×F = 8 × 1 = 8 C = 3×F = 3 × 1 = 3 D = 4×F = 4 × 1 = 4 E = 2×F = 2 × 1 = 2 and ABCDEF = 783421