What is the prime factorization of 189?
Understand the Problem
The question is asking for the prime factorization of the number 189. This means we need to determine the prime numbers that multiply together to give 189.
Answer
The prime factorization of 189 is \( 3^3 \times 7 \).
Answer for screen readers
The prime factorization of 189 is ( 3^3 \times 7 ).
Steps to Solve
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Start with the number We begin with the number 189.
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Check for divisibility by the smallest prime We check if 189 is divisible by 2 (the smallest prime number). Since 189 is odd, it is not divisible by 2.
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Check divisibility by the next prime (3) Next, we check if 189 is divisible by 3. To do this, we sum the digits of 189:
$$ 1 + 8 + 9 = 18 $$
Since 18 is divisible by 3, 189 is also divisible by 3. We divide:
$$ 189 \div 3 = 63 $$ -
Continue factoring 63 Now we factor 63. Again, we check for divisibility by 3. The sum of the digits of 63 is:
$$ 6 + 3 = 9 $$
Since 9 is divisible by 3, we divide:
$$ 63 \div 3 = 21 $$ -
Factor 21 Next, we factor 21. The sum of digits is:
$$ 2 + 1 = 3 $$
Since 3 is divisible by 3, we divide:
$$ 21 \div 3 = 7 $$ -
Identify the last factor The last number we have is 7, which is a prime number. Therefore, we cannot factor it further.
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Combine all prime factors Now, we can combine all the prime factors obtained from our divisions. We have:
$$ 189 = 3 \times 3 \times 3 \times 7 $$
This can also be written using exponentiation as:
$$ 189 = 3^3 \times 7 $$
The prime factorization of 189 is ( 3^3 \times 7 ).
More Information
The prime factorization is helpful in various applications, such as finding the greatest common divisor (GCD) or least common multiple (LCM), as well as in simplifying fractions.
Tips
One common mistake is forgetting to check divisibility by all potential prime factors or stopping the factorization too early. Be sure to check beyond the first few primes as factors.
