Find the greatest number which divides 65 and 117 completely.

Step 1: Understand the problem

The problem asks for the greatest number that divides both 65 and 117 completely. This is equivalent to finding the Greatest Common Divisor (GCD) of 65 and 117.

Step 2: Find prime factors of 65

To find the GCD, we first find the prime factorization of each number. For 65, we can see that it is divisible by 5, and $65 \div 5 = 13$. Since 13 is a prime number, the prime factorization of 65 is $5 \times 13$.

Step 3: Find prime factors of 117

Next, we find the prime factorization of 117. We can see that the sum of its digits ($1+1+7=9$) is divisible by 3, so 117 is divisible by 3. $117 \div 3 = 39$. Again, 39 is divisible by 3, so $39 \div 3 = 13$. Since 13 is a prime number, the prime factorization of 117 is $3 \times 3 \times 13$, or $3^2 \times 13$.

Step 4: Identify common prime factors

Now we compare the prime factorizations of both numbers: $65 = 5 \times 13$ and $117 = 3^2 \times 13$. The only common prime factor is 13.

Step 5: Calculate the GCD

The Greatest Common Divisor is the product of the common prime factors raised to the lowest power they appear in either factorization. In this case, the only common prime factor is 13, and it appears with a power of 1 in both factorizations. Therefore, the GCD of 65 and 117 is 13.

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