Puzzle for August 3, 2021 ( )
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 D with A + B (from eq.3): F = A + A + B which becomes eq.5a) F = 2×A + B
Hint #2
In eq.4, replace F with 2×A + B (from eq.5a): E = A + 2×A + B which becomes eq.4a) E = 3×A + B
Hint #3
In eq.6, replace E with 3×A + B (from eq.4a): B – A = A – B + 3×A + B which becomes B – A = 4×A Add A to both sides of the above equation: B – A + A = 4×A + A which makes B = 5×A
Hint #4
In eq.3, substitute 5×A for B: D = A + 5×A which makes D = 6×A
Hint #5
In eq.4a, substitute 5×A for B: E = 3×A + 5×A which makes E = 8×A
Hint #6
In eq.5a, substitute 5×A for B: F = 2×A + 5×A which makes F = 7×A
Hint #7
Substitute 7×A for F, and 5×A for B in eq.2: C = 7×A – 5×A which makes C = 2×A
Solution
Substitute 5×A for B, 2×A for C, 6×A for D, 8×A for E, and 7×A for F in eq.1: A + 5×A + 2×A + 6×A + 8×A + 7×A = 29 which simplifies to 29×A = 29 Divide both sides of the above equation by 29: 29×A ÷ 29 = 29 ÷ 29 which means A = 1 making B = 5×A = 5 × 1 = 5 C = 2×A = 2 × 1 = 2 D = 6×A = 6 × 1 = 6 E = 8×A = 8 × 1 = 8 F = 7×A = 7 × 1 = 7 and ABCDEF = 152687