Puzzle for June 21, 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 E with C + F (from eq.4): D + F = C + F – F which becomes eq.5a) D + F = C
Hint #2
In eq.2, replace D + F with C (from eq.5a): B = C
Hint #3
In eq.6, replace C with D + F (from eq.5a): D + F – A = A + F In the above equation, subtract F from both sides, and add A to both sides: D + F – A – F + A = A + F – F + A which simplifies to D = 2×A
Hint #4
In eq.3, substitute 2×A for D: F = A + 2×A which makes F = 3×A
Hint #5
Substitute 2×A for D, and 3×A for F in eq.2: B = 2×A + 3×A which makes B = 5×A and also makes C = B = 5×A
Hint #6
Substitute 5×A for C, and 3×A for F in eq.4: E = 5×A + 3×A which makes E = 8×A
Solution
Substitute 5×A for B and C, 2×A for D, 8×A for E, and 3×A for F in eq.1: A + 5×A + 5×A + 2×A + 8×A + 3×A = 24 which simplifies to 24×A = 24 Divide both sides of the above equation by 24: 24×A ÷ 24 = 24 ÷ 24 which means A = 1 making B = C = 5×A = 5 × 1 = 5 D = 2×A = 2 × 1 = 2 E = 8×A = 8 × 1 = 8 F = 3×A = 3 × 1 = 3 and ABCDEF = 155283