Puzzle for September 16, 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.6, replace D + F with A + B (from eq.5): C + E = B + A + B which becomes eq.6a) C + E = A + 2×B
Hint #2
Add E to both sides of eq.2: B + C + E = A + E In the above equation, replace C + E with A + 2×B (from eq.6a): B + A + 2×B = A + E which becomes A + 3×B = A + E Subtract A from both sides: A + 3×B – A = A + E – A which makes 3×B = E
Hint #3
In eq.3, substitute 3×B for E: eq.3a) 3×B + F = A
Hint #4
Substitute 3×B + F for A (from eq.3a) into eq.5: D + F = 3×B + F + B which becomes D + F = 4×B + F Subtract F from each side of the equation above: D + F – F = 4×B + F – F which makes D = 4×B
Hint #5
Substitute 4×B for D in eq.4: F = B + 4×B which makes F = 5×B
Hint #6
Substitute 3×B for E, and 5×B for F in eq.3: 3×B + 5×B = A which makes 8×B = A
Hint #7
Substitute 8×B for A in eq.2: B + C = 8×B Subtract B from each side of the above equation: B + C – B = 8×B – B which makes C = 7×B
Solution
Substitute 8×B for A, 7×B for C, 4×B for D, 3×B for E, and 5×B for F in eq.3: 8×B + B + 7×B + 4×B + 3×B + 5×B = 28 which simplifies to 28×B = 28 Divide both sides of the above equation by 28: 28×B ÷ 28 = 28 ÷ 28 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 7×B = 7 × 1 = 7 D = 4×B = 4 × 1 = 4 E = 3×B = 3 × 1 = 3 F = 5×B = 5 × 1 = 5 and ABCDEF = 817435