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7k - 4 \leq 50
7k \leq 54
k \leq \frac{54}{7} \approx 7.71
Thus, \( k \) can be any integer from 1 to 7. Therefore, there are 7 integers in the first 50 that are congruent to 3 modulo 7. The answer is \(\boxed{7}\).
Question:** A pharmacologist developing new drugs notes that effective doses often follow a pattern related to prime numbers and divisibility, reflecting nature’s own balance. What is the sum of all the odd divisors of 180?
First, factorize 180:
= 2^2 \times 3^2 \times 5
The odd divisors are those not divisible by 2, so we consider only the odd part: \( 3^2 \times 5 \).
The odd divisors are all numbers of the form \( 3^a \times 5^b \) where \( 0 \leq a \leq 2 \), \( 0 \leq b \leq 1 \).
List them: