Question

In a number system, the product of 44 and 11 is 3414. The number 3111 of this system, when converted to the decimal number system, becomes:

• A. 406
• B. 1086
• C. 213
• D. 691

Hint:

Convert each digit position using powers of the base to find the decimal equivalent.

Answer 1

The Correct Option is B

Let base = $b$. $44_b = 4b + 4$, $11_b = b + 1$. Their product = $(4b+4)(b+1) = 4b^2 + 8b + 4$. In base $b$, $3414_b = 3b^3 + 4b^2 + b + 4$.
Equating: $4b^2 + 8b + 4 = 3b^3 + 4b^2 + b + 4 \Rightarrow 3b^3 - 7b = 0 \Rightarrow b(b- \sqrt[?]{})$, solving gives $b=6$. Then $3111_6 = 3(216) + 1(36) + 1(6) + 1 = 648 + 36 + 6 + 1 = 691$. Correction: Correct Answer = (4) 691.