Puzzle for June 11, 2019 ( )
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.2: D - E - F + F = B + F which becomes D - E = B + F In eq.6, replace D - E with B + F: A + B = C + B + F Subtract B from both sides: A + B - B = C + B + F - B which becomes eq.2a) A = C + F
Hint #2
In eq.4, replace A with C + F (from eq.2a): E - F = C + F - C which becomes E - F = F Add F to both sides of the above equation: E - F + F = F + F which makes E = 2×F
Hint #3
In eq.3, substitute C + F for A (from eq.2a): B + C = C + F + E Subtract C from both sides of the above equation: B + C - C = C + F + E - C which becomes B = F + E Substitute 2×F for E: B = F + 2×F which makes B = 3×F
Hint #4
Substitute 2×F for E, and 3×F for B in eq.2: D - 2×F - F = 3×F which becomes D - 3×F = 3×F Add 3×F to both sides of the above equation: D - 3×F + 3×F = 3×F + 3×F which makes D = 6×F
Hint #5
Substitute 2×F for E, 3×F for B, and 6×F for D in eq.5: C + 2×F = 3×F + 6×F Subtract 2×F from both sides: C + 2×F - 2×F = 3×F + 6×F - 2×F which makes C = 7×F
Hint #6
Substitute 7×F for C in eq.2a: A = 7×F + F which makes A = 8×F
Solution
Substitute 8×F for A, 3×F for B, 7×F for C, 6×F for D, and 2×F for E in eq.1: 8×F + 3×F + 7×F + 6×F + 2×F + F = 27 which simplifies to 27×F = 27 Divide both sides of the above equation by 27: 27×F ÷ 27 = 27 ÷ 27 which means F = 1 making A = 8×F = 8 × 1 = 8 B = 3×F = 3 × 1 = 3 C = 7×F = 7 × 1 = 7 D = 6×F = 6 × 1 = 6 E = 2×F = 2 × 1 = 2 and ABCDEF = 837621