Puzzle for July 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.5, replace A + B with E (from eq.3): eq.5a) F = E + C
Hint #2
In eq.4, replace F with E + C (from eq.5a): E + C = A + E Subtract E from each side of the equation above: E + C – E = A + E – E which makes C = A
Hint #3
In eq.2, substitute A for C: D = A + A which makes D = 2×A
Hint #4
Substitute A for C in eq.5: F = A + B + A which becomes eq.5b) F = 2×A + B
Hint #5
Substitute 2×A for D, A + B for E (from eq.3), A for C, and 2×A + B for F (from eq.5b) in eq.6: B – 2×A + A + B = A + 2×A + 2×A + B which becomes 2×B – A = 5×A + B In the equation above, add A to both sides, and subtract B from both sides: 2×B – A + A – B = 5×A + B + A – B which makes B = 6×A
Hint #6
Substitute 6×A for B in eq.3: E = A + 6×A which makes E = 7×A
Hint #7
Substitute 6×A for B in eq.5b: F = 2×A + 6×A which makes F = 8×A
Solution
Substitute 6×A for B, A for C, 2×A for D, 7×A for E, and 8×A for F in eq.1: A + 6×A + A + 2×A + 7×A + 8×A = 25 which simplifies to 25×A = 25 Divide both sides of the above equation by 25: 25×A ÷ 25 = 25 ÷ 25 which means A = 1 making B = 6×A = 6 × 1 = 6 C = A = 1 D = 2×A = 2 × 1 = 2 E = 7×A = 7 × 1 = 7 F = 8×A = 8 × 1 = 8 and ABCDEF = 161278