To find the greatest common divisor (GCD) of 10 and 315, we use the following steps:
1. Understand the Definition: The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder.
2. Prime Factorization of Each Number:
- Prime factorization helps in identifying common factors.
- The prime factors of 10 are: [tex]\( 10 = 2 \times 5 \)[/tex].
- The prime factors of 315 are: [tex]\( 315 = 3^2 \times 5 \times 7 \)[/tex].
3. Identify Common Factors:
- From the factorizations, the common prime factors are identified.
- The common factor between the prime factorizations of 10 and 315 is 5.
4. Determine the GCD:
- Since 5 is the largest common factor from the prime factorizations of both numbers, we conclude that the greatest common divisor is 5.
Hence, the greatest common divisor (GCD) of 10 and 315 is [tex]\( \boxed{5} \)[/tex].
