+2653 How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?
+17 To find the bases b≥2 such that $100_b + 1_b is prime, we need to understand how to convert numbers from different bases to base 10 and perform the addition.
1.Convert 100b to base 10: 100_b = 1 * b^2 + 0 * b^1 + 0 * b^0 = b^22.Convert 1b to base 10: 1_b = 1 * b^0 = 13. Add the two numbers in base 10: b^2 + 1 For the sum to be prime, b^2 + 1 should be a prime number. Let′s consider some examples to understand this further: −Ifb = 2, then 22+1=5, which is prime. - If b=3, then $3^2 + 1 = 10,whichisnotprime.−Ifb = 4,then42+1=17, which is prime. Therefore, the bases b≥2 such that 100_b + 1_b is prime are 2 and 4.
