91Ó°ÊÓ

To break down a number into its prime factors, you can use divisibility rules to simplify the process. Divisibility rules are shortcuts that help determine whether a number can be divided by another number without a remainder.

For example, in our problem with 693, we start by checking divisibility by 2. Since 693 is odd, it is not divisible by 2.

We then move to the rule for 3: a number is divisible by 3 if the sum of its digits is divisible by 3. Adding 6 + 9 + 3 gives 18, which is divisible by 3. This tells us that 693 can be divided by 3.

Applying these rules makes it faster to find factors without performing complicated divisions. These are just a couple of examples, and similar rules exist for numbers like 5, 7, and 11.

Prime numbers are numbers that have only two distinct factors: 1 and themselves. This means they cannot be divided by any other number without leaving a remainder.

In our example, after breaking down 693 using the divisibility rules for 3 and 7, we arrive at the number 11. A quick check will show that 11 cannot be divided by any number other than 1 and 11 itself. This confirms that 11 is a prime number.

Understanding prime numbers is essential because they are the building blocks of all numbers. When you factorize a number, the end goal is to express it as a product of prime numbers.

Factorization is the process of breaking down a number into smaller numbers that, when multiplied together, give the original number. Specifically, prime factorization involves breaking the number down until all the factors are prime numbers.

Let's look at 693 one more time. We start by using the divisibility rule for 3 and get 231. Factoring 231 by 3 again, we get 77. Finally, we divide 77 by 7 to get 11, which is a prime number.

The complete factorization of 693 is then:

• 693 ÷ 3 = 231
• 231 ÷ 3 = 77
• 77 ÷ 7 = 11

The factors are 3, 3, 7, and 11.

So, the prime factorization of 693 is: \(693 = 3^2 \times 7 \times 11\)

Mastering factorization helps significantly in simplifying problems in algebra and number theory.