Puzzle for September 2, 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
In eq.2, replace D + E with A (from eq.5): C – A = A + A Add A to both sides of the above equation: C – A + A = A + A + A which makes C = 3×A
Hint #2
In eq.6, replace C with 3×A: F = 3×A – A which means F = 2×A
Hint #3
Substitute 2×A for F in eq.4: E = 2×A ÷ A which makes E = 2
Hint #4
Substitute 2 for E in eq.5: D + 2 = A Subtract 2 from both sides of the above equation: D + 2 – 2 = A – 2 which makes eq.5a) D = A – 2
Hint #5
Substitute 3×A for C, and 2×A for F in eq.3: B + 3×A = A + 2×A which becomes B + 3×A = 3×A Subtract 3×A from both sides of the above equation: B + 3×A – 3×A = 3×A – 3×A which makes B = 0
Solution
Substitute 0 for B, 3×A for C, A – 2 for D (from eq.5a), 2 for E, and 2×A for F in eq.1: A + 0 + 3×A + A – 2 + 2 + 2×A = 21 which simplifies to 7×A = 21 Divide both sides of the equation above by 7: 7×A ÷ 7 = 21 ÷ 7 which means A = 3 making C = 3×A = 3 × 3 = 9 D = A – 2 = 3 – 2 = 1 (from eq.5a) F = 2×A = 2 × 3 = 6 and ABCDEF = 309126