Puzzle for June 12, 2023 ( )
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.5, replace E with B + C (from eq.4): B + C - F = C + F In the above equation, subtract C from both sides, and add F to both sides: B + C - F - C + F = C + F - C + F which makes B = 2×F
Hint #2
In eq.6, replace B with 2×F: F - 2×F = 2×F - C which becomes -F = 2×F - C Add F and C to both sides of the above equation: -F + F + C = 2×F - C + F + C which makes C = 3×F
Hint #3
In eq.5, substitute 3×F for C: E - F = 3×F + F which becomes E - F = 4×F Add F to both sides of the equation above: E - F + F = 4×F + F which makes E = 5×F
Hint #4
Substitute 3×F for C in eq.2: D - 3×F = 3×F - F which becomes D - 3×F = 2×F Add 3×F to both sides of the above equation: D - 3×F + 3×F = 2×F + 3×F which makes D = 5×F
Hint #5
Substitute 5×F for D in eq.3: A = 5×F + F which makes A = 6×F
Solution
Substitute 6×F for A, 2×F for B, 3×F for C, and 5×F for D and E in eq.1: 6×F + 2×F + 3×F + 5×F + 5×F + F = 22 which simplifies to 22×F = 22 Divide both sides of the above equation by 22: 22×F ÷ 22 = 22 ÷ 22 which means F = 1 making A = 6×F = 6 × 1 = 6 B = 2×F = 2 × 1 = 2 C = 3×F = 3 × 1 = 3 D = E = 5×F = 5 × 1 = 5 and ABCDEF = 623551