Puzzle for July 6, 2020 ( )
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.4, replace C with A + D (from eq.2): A + D – D – F = A + B which becomes A – F = A + B Subtract A from each side of the equation above: A – F – A = A + B – A which simplifies to –F = B Since B and F must be non–negative integers, the above equation makes B = 0 and F = 0
Hint #2
In eq.5, replace B and F with 0, and C with A + D (from eq.2): D + 0 = (A + D – D + 0) ÷ A which simplifies to D = (A) ÷ A which means D = 1
Hint #3
In eq.2, substitute 1 for D: eq.2a) C = A + 1
Hint #4
Substitute A + 1 for C (from eq.2a) in eq.3: E = A + A + 1 which means eq.3a) E = 2×A + 1
Solution
Substitute 0 for B and F, A + 1 for C (from eq.2a), 1 for D, and 2×A + 1 for E (from eq.3a) in eq.1: A + 0 + A + 1 + 1 + 2×A + 1 + 0 = 15 which simplifies to 4×A + 3 = 15 Subtract 3 from both sides of the equation above: 4×A + 3 – 3 = 15 – 3 which makes 4×A = 12 Divide both sides by 4: 4×A ÷ 4 = 12 ÷ 4 which means A = 3 making C = A + 1 = 3 + 1 = 4 (from eq.2a) E = 2×A + 1 = 2×3 + 1 = 6 + 1 = 7 (from eq.3a) and ABCDEF = 304170