The decimal number 61 is a 2-digit number. Find the smallest positive integer B so that when we express the decimal number 61 as a base B number we still get a 2-digit number.
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Let's start by expressing the decimal number 61 as a base B number. In general, any positive integer can be expressed as the sum of powers of its base, so we can write:
61 = a0 + a1B^1 + a2B^2 + ... + ak*B^k
where ai are digits in base B, with ai < B for all i, and k is the highest power of B that is less than or equal to 61.
Since we want the base B number to be a 2-digit number, we know that 10 ≤ 61 < B^2, so we can set k = 1 and try to solve for B:
61 = a0 + a1*B^1
Since 61 is a prime number, it can only be expressed as a sum of 1 and itself. Therefore, we have:
a1 = 6 and a0 = 1
Substituting these values into the previous equation, we get:
61 = 1 + 6B
Solving for B, we get:
B = (61 - 1) / 6 = 10
Therefore, the smallest positive integer B such that the base B representation of 61 is a 2-digit number is B = 10.