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Chapter 1: Problem 82

Factor the given number into its prime factors. If the number is prime, say so. $$1,332$$

Short Answer

Expert verified
The prime factorization of 1,332 is 2^2 脳 3^2 脳 37.

Step by step solution

01

Check divisibility by 2

The number 1,332 is even, so it is divisible by 2. Divide 1,332 by 2 to get 666. Therefore, 1,332 = 2 脳 666.
02

Check divisibility of 666 by 2

The number 666 is also even, so it is divisible by 2. Divide 666 by 2 to get 333. Therefore, 1,332 = 2 脳 2 脳 333.
03

Check divisibility of 333 by 3

The sum of the digits of 333 (which is 3 + 3 + 3 = 9) is divisible by 3, so 333 is divisible by 3. Divide 333 by 3 to get 111. Therefore, 1,332 = 2 脳 2 脳 3 脳 111.
04

Check divisibility of 111 by 3

The sum of the digits of 111 (which is 1 + 1 + 1 = 3) is divisible by 3, so 111 is divisible by 3. Divide 111 by 3 to get 37. Therefore, 1,332 = 2 脳 2 脳 3 脳 3 脳 37.
05

Determine if 37 is prime

Check if 37 is a prime number. 37 is not divisible by any number other than 1 and itself, so it is a prime number. Therefore, we cannot factor 37 any further.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

divisibility rules
Divisibility rules are super handy when determining if a number can be divided by another without a remainder. They save time and can be applied mentally, making complex math easier.
Here are some key rules:
  • Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
  • Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For example, for 333, the sum is 9 (3 + 3 + 3).
  • Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
  • Divisibility by 7, 11, 13, etc.: These rules get more complicated, but often checking smaller primes first simplifies factorization.
Using these rules, you can test for divisibility step-by-step to break down a complex number like 1,332 efficiently.
prime numbers
Prime numbers are fascinating and fundamental in mathematics. A prime number is a natural number greater than 1, which has no positive divisors other than 1 and itself.
For example, 2, 3, 5, 7, and 11 are prime numbers. They cannot be divided evenly by any other numbers.
  • 2 is the smallest and only even prime number.
  • 37, as found in the solution, is a prime because it cannot be divided by any number other than 1 and 37 itself.
  • Prime numbers are the building blocks of all natural numbers through multiplication.
Knowing if a number is prime helps determine when to stop the factorization process, as primes cannot be broken down further.
factorization steps
Factorization is the process of breaking down a number into its prime factors until what remains are all prime numbers.
Here鈥檚 how we factorize 1,332 step-by-step using prime numbers:
  1. We start by checking if 1,332 is divisible by the smallest prime, which is 2, since it's even. Dividing, we get 666.
  2. Next, we check 666 for divisibility by 2 again, since it's even. Dividing, we get 333.
  3. For 333, we check divisibility by 3. Adding the digits (3 + 3 + 3 = 9), confirms it. Dividing, we get 111.
  4. We take 111 and use the sum rule for 3 (1 + 1 + 1 = 3). It's divisible, giving 37.
  5. Finally, 37 is checked for primality and found to be prime.
Thus, 1,332 can be expressed as the product of its prime factors: \(2 \times 2 \times 3 \times 3 \times 37\)
This systematic approach ensures that every factor is prime and the factorization is complete.

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