Question

what is the smallest whole number $b$ such that $62$ can be expressed in base $b$ using only three digits?

Answer 1

To find the smallest whole number b such that 62 can be expressed in base b using only three digits, we need to solve a quadratic equation. By setting the expression for 62 equal to 6b^2 + 2b, we can rearrange the equation and use the quadratic formula to find the values of b. The smallest whole number b that satisfies the conditions is 7.

Step-by-step explanation:

To find the smallest whole number b such that 62 can be expressed in base b using only three digits, we can start by considering the largest possible digit in base b. Since we want only three digits, the largest digit will be one less than the base itself. So, we have b - 1 as the largest digit. Now, we need to check which value of b will result in a representation for 62 using only three digits. In base b, the representation of 62 will be 6b^2 + 2b + 0. Setting this expression equal to 62, we have 6b^2 + 2b = 62. Rearranging the equation, we get 6b^2 + 2b - 62 = 0. This is a quadratic equation that we can solve to find the values of b.Using the quadratic formula, b is given by b = (-2 ± √(2^2 - 4(6)(-62))) / (2(6)). Calculating the values, we get b ≈ 6.41 and b ≈ -1.57. Since we are looking for the smallest whole number, the answer is b = 7. Therefore, the smallest whole number b such that 62 can be expressed in base b using only three digits is 7.

Answer 2

The smallest whole number b such that 62 can be expressed in base b with three digits is 7. The number 62 in base 7 is represented as 120.

Step-by-step explanation:

The student is asking for the smallest whole number b so that the number 62 can be represented as a three-digit number in base b. To find this, we have to express 62 in base b such that it fits the format of b^2 + b^1 + b^0, where each exponent represents a place value (hundreds, tens, and ones respectively) in this new base system. Since we are looking for a three-digit number, the base b must be such that b^2 is the largest square smaller than or equal to 62.

Trying the bases one by one: in base 8, 64 is the largest square (8^2), but that exceeds 62. So we consider base 7. The largest square in base 7 is 49 (7^2), which is less than 62. 62 in base 7 is (1×7^2) + (2×7^1) + (0×7^0), which translates to 120 in base 7. Therefore, the smallest whole number b is 7, and the representation of 62 in base 7 using three digits is 120.