360 is divisible by which of the following numbers 2,3,4,5,6,9, or 10
The Correct Answer and Explanation is :
To determine which of the numbers 2, 3, 4, 5, 6, 9, and 10 are divisors of 360, we need to perform a divisibility test for each number.
Step 1: Prime Factorization of 360
First, let’s find the prime factorization of 360 to help us understand its divisibility:
- Start by dividing by the smallest prime number, 2:
- (360 \div 2 = 180)
- (180 \div 2 = 90)
- (90 \div 2 = 45)
- Next, we divide by the next smallest prime, which is 3:
- (45 \div 3 = 15)
- (15 \div 3 = 5)
- Finally, we are left with the prime number 5:
- (5 \div 5 = 1)
Thus, the prime factorization of 360 is:
[ 360 = 2^3 \times 3^2 \times 5^1 ]
Step 2: Divisibility Tests
Now, we can check each number for divisibility based on the factorization:
- 2: Since 360 includes (2^3), it is divisible by 2.
- 3: The factorization includes (3^2), so 360 is divisible by 3.
- 4: 4 can be expressed as (2^2). Since 360 includes (2^3), it is divisible by 4.
- 5: The factorization includes (5^1), so 360 is divisible by 5.
- 6: Since 6 is the product of 2 and 3, and both factors are present in the factorization, 360 is divisible by 6.
- 9: 9 is (3^2), and since 360 includes (3^2), it is divisible by 9.
- 10: 10 is the product of 2 and 5, both of which are present, so 360 is divisible by 10.
Conclusion
In summary, 360 is divisible by all of the following numbers: 2, 3, 4, 5, 6, 9, and 10. This shows the importance of understanding prime factorization, as it simplifies the process of determining divisibility by breaking down the numbers into their fundamental components.