Question: In a number system the product of 44 and 11 is 1034. The number 3111 of this system, when converted to the decimal number system, becomes
Explanation:
Let the base be b.
∴ (4b + 4)(b + 1) = b3 + 3b + 4
∴ b3 − 4b2 − 5b = 0
Solving we get, b = 0, –1, 5
∵ Base cannot be zero or negative, so base is 5.
∴ (3111)5 = 3 × 125 + 25 + 5 + 1 = 406
Hence, option (a).
Alternatively,
Let the required base be x.
We know that the answer of 44 × 11 in the required base is 1034.
∴ As 4 occurs in the product, we can say that the base is greater than 4.
Also, 44 × 11 = 484 in base 10.
As 1034 > 484, the base is lesser than 10.
So we can represent the multiplication as follows:
44 × 11 = 44(x + 1) = 44x + 44 = 1034
4 + 4 → 3 or 13 or 23 …
As the base is less than 10, 4 + 4, which is 8 in base 10 cannot be expressed as 3 in the required base.
∴ 4 + 4 → 13 or 23…
4 + 4 → 13
∴ 8 → 13
∴ 8 → x + 3
∴ x = 5
If 4 + 4 → 23
∴ 8 → 23
∴ 8 → x + 13
∴ x = –5, which is not possible.
∴ (3111)5 = 1 × 50 + 1 × 51 + 1 × 52 + 3 × 53 = (406)10
Hence, option (a).