Find the greatest number which divides 65 and 117 completely.
Get the complete, step-by-step math solution for: "Find the greatest number which divides 65 and 117 completely.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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÷5=13. Since 13 is a prime number, the prime factorization of 65 is 5×13.
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÷3=39. Again, 39 is divisible by 3, so 39÷3=13. Since 13 is a prime number, the prime factorization of 117 is 3×3×13, or 32×13.
Step 4: Identify common prime factors
Now we compare the prime factorizations of both numbers: 65=5×13 and 117=32×13. 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.