Final answer:
Binary number 11111₂ converts to decimal as 31, and then 31 converts to quinary (base-5) number system as 111₅.
Step-by-step explanation:
To convert the binary number 111112 into the quinary (base-5) numeration system, we have to first understand its value in decimal, as it is easier to convert from decimal to other bases. The binary number 111112 can be evaluated as follows:
- (1 × 24) + (1 × 23) + (1 × 22) + (1 × 21) + (1 × 20)
- 16 + 8 + 4 + 2 + 1
- 3110
To then convert 31 into the quinary system, we divide by the largest powers of 5 and take the remainders as coefficients:
- 31 ÷ 52 (25) = 1 with a remainder of 6
- 6 ÷ 51 (5) = 1 with a remainder of 1
- 1 ÷ 50 (1) = 1 with no remainder
Putting the coefficients together gives us 1115. Therefore, the binary number 111112 is equal to the quinary number 1115.