Question

What is 874 base 9 multiplied by 676 base 9?​

Answer 1

Consider the digit expansion of one of the numbers, say,

676₉ = 600₉ + 70₉ + 6₉

then distribute 874₉ over this sum.

874₉ • 6₉ = (8•6)(7•6)(4•6)₉ = (48)(42)(24)₉

• 48 = 45 + 3 = 5•9¹ + 3•9⁰ = 53₉

• 42 = 36 + 6 = 4•9¹ + 6•9⁰ = 46₉

• 24 = 18 + 6 = 2•9¹ + 6•9⁰ = 26₉

874₉ • 6₉ = 5(3 + 4)(6 + 2)6₉ = 5786₉

874₉ • 70₉ = (8•7)(7•7)(4•7)0₉ = (56)(49)(28)0₉

• 56 = 54 + 2 = 6•9¹ + 2•9⁰ = 62₉

• 49 = 45 + 4 = 5•9¹ + 4•9⁰ = 54₉

• 28 = 27 + 1 = 3•9¹ + 1•9⁰ = 31₉

874₉ • 70₉ = 6(2 + 5)(4 + 3)10₉ = 67710₉

874₉ • 600₉ = (874•6)00₉ = 578600₉

Then

874₉ • 676₉ = 578600₉ + 67710₉ + 5786₉

= 5(7 + 6)(8 + 7 + 5)(6 + 7 + 7)(0 + 1 + 8)(0 + 0 + 6)₉

= 5(13)(20)(20)(1•9)6₉

= 5(13)(20)(20 + 1)06₉

= 5(13)(20)(2•9 + 3)06₉

= 5(13)(20 + 2)306₉

= 5(13)(2•9 + 4)306₉

= 5(13 + 2)4306₉

= 5(1•9 + 6)4306₉

= (5 + 1)64306₉

= 664306₉