Is 465 a Prime Number?

There are two main categories of numbers — prime numbers and composite numbers, depending on the number of factors they have.

A prime number is a natural number that is divisible only by 1 and itself. For example, 3 is a prime number because it is divisible by 1 and itself.

A composite number is a positive number that is divisible by more than two numbers. For example, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers have a few properties, such as:

Prime numbers are positive numbers always greater than 1.

2 is the only even prime number.

They have only two factors: 1 and the number itself.

Any two distinct prime numbers are co-prime numbers because they have only one common factor, which is 1.

As 465 has more than two factors, it is not a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 465 has more than two factors, it is not a prime number. Several methods are used to distinguish between prime and composite numbers, including:

• Counting Divisors Method

• Divisibility Test

• Prime Number Chart

• Prime Factorization

The method in which we count the number of divisors to categorize the numbers as prime or composite is called the counting divisors method. Based on the count of the divisors, we categorize numbers as prime or composite.

If there is a total count of only 2 divisors, then the number would be prime.

If the count is more than 2, then the number is composite. Let’s check whether 465 is prime or composite.

Step 1: All numbers are divisible by 1 and itself.

Step 2: Divide 465 by 2. It is not divisible by 2, so 2 is not a factor of 465.

Step 3: Divide 465 by 3. The sum of the digits is 15, which is divisible by 3, so 3 is a factor of 465.

Step 4: You can simplify checking divisors up to 465 by finding the root value. We then need to only check divisors up to the root value.

Step 5: When we divide 465 by 5, 9, and 15, it is divisible by 5 and 15. Since 465 has more than 2 divisors, it is a composite number.

We use a set of rules to check whether a number is divisible by another number completely or not. This is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 5. Since 5 is an odd number, 465 is not divisible by 2. -

Divisibility by 3: The sum of the digits in the number 465 is 15. Since 15 is divisible by 3, 465 is divisible by 3.

Divisibility by 5: The unit’s place digit is 5. Therefore, 465 is divisible by 5.

Divisibility by 7: The last digit in 465 is 5. To check divisibility by 7, double the last digit (5 × 2 = 10). Then, subtract it from the rest of the number (46 - 10 = 36). Since 36 is not divisible by 7, 465 is not divisible by 7.

Divisibility by 11: In 465, the alternating sum of the digits is 3 - 6 + 5 = 2, which is not divisible by 11. Since 465 is divisible by 3 and 5, it has more than two factors.

Therefore, it is a composite number.

The prime number chart is a tool created using a method called “The Sieve of Eratosthenes.” In this method, we follow the steps below:

Step 1: Write numbers from 1 to 100 in 10 rows and 10 columns.

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.

Step 3: Mark 2 because it is a prime number and cross out all the multiples of 2.

Step 4: Mark 3 because it is a prime number and cross out all the multiples of 3.

Step 5: Repeat this process until you reach the table consisting of marked and crossed boxes, except 1. Through this process, we will have a list of prime numbers from 1 to 100. The list is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. 465 is not present in the list of prime numbers, so it is a composite number.

Prime factorization is a process of breaking down a number into prime factors, then multiplying those factors to obtain the original number.

Step 1: We can write 465 as 5 × 93.

Step 2: In 5 × 93, 93 is a composite number. Further, break down 93 into 3 × 31.

Step 3: Now we get the product consisting of only prime numbers. Hence, the prime factorization of 465 is 3 × 5 × 31.

• Prime number: A natural number greater than 1 that has no positive divisors other than 1 and itself.

• Composite number: A natural number greater than 1 that has more than two positive divisors.

• Divisibility: The ability for one number to be divided by another without leaving a remainder.

• Prime factorization: Expressing a number as a product of its prime factors.

• Factors: Numbers that can be multiplied together to get another number. For example, the factors of 6 are 1, 2, 3, and 6.