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Answer :
Sure, I can help you understand the prime factorization of the given numbers by breaking it down step-by-step:
1. Prime Factorization of 252:
- Start by dividing 252 by the smallest prime, 2:
- [tex]\(252 \div 2 = 126\)[/tex].
- Divide 126 by 2:
- [tex]\(126 \div 2 = 63\)[/tex].
- 63 is not divisible by 2, so move to the next smallest prime, 3:
- [tex]\(63 \div 3 = 21\)[/tex].
- Divide 21 by 3:
- [tex]\(21 \div 3 = 7\)[/tex].
- 7 is a prime number, so we stop here.
- Thus, the prime factorization of 252 is [tex]\(2^2 \times 3^2 \times 7\)[/tex].
2. Prime Factorization of 360:
- Start with 360 and divide by 2:
- [tex]\(360 \div 2 = 180\)[/tex].
- Divide 180 by 2:
- [tex]\(180 \div 2 = 90\)[/tex].
- Divide 90 by 2:
- [tex]\(90 \div 2 = 45\)[/tex].
- 45 is not divisible by 2, move to 3:
- [tex]\(45 \div 3 = 15\)[/tex].
- Divide 15 by 3:
- [tex]\(15 \div 3 = 5\)[/tex].
- 5 is a prime number.
- So, the prime factorization of 360 is [tex]\(2^3 \times 3^2 \times 5\)[/tex].
3. Prime Factorization of 848:
- Divide 848 by 2:
- [tex]\(848 \div 2 = 424\)[/tex].
- Divide 424 by 2:
- [tex]\(424 \div 2 = 212\)[/tex].
- Divide 212 by 2:
- [tex]\(212 \div 2 = 106\)[/tex].
- Divide 106 by 2:
- [tex]\(106 \div 2 = 53\)[/tex].
- 53 is a prime number.
- Therefore, the prime factorization of 848 is [tex]\(2^4 \times 53\)[/tex].
4. Prime Factorization of 2890:
- Divide 2890 by 2:
- [tex]\(2890 \div 2 = 1445\)[/tex].
- 1445 is not divisible by 2, check with 5:
- [tex]\(1445 \div 5 = 289\)[/tex].
- 289 is not divisible by 5, check with other primes (it's actually [tex]\(17^2\)[/tex]).
- Prime factorization of 2890 is [tex]\(2 \times 5 \times 17^2\)[/tex].
5. Prime Factorization of 28910:
- Divide 28910 by 2:
- [tex]\(28910 \div 2 = 14455\)[/tex].
- 14455 is divisible by 5:
- [tex]\(14455 \div 5 = 2891\)[/tex].
- 2891 is divisible by 7:
- [tex]\(2891 \div 7 = 413\)[/tex].
- Finally, divide 413 by the next prime:
- [tex]\(413 \div 59 = 7\)[/tex].
- 7 is a prime.
- Prime factorization of 28910 is [tex]\(2 \times 5 \times 7^2 \times 59\)[/tex].
6. Prime Factorization of 234750:
- Divide 234750 by 2:
- [tex]\(234750 \div 2 = 117375\)[/tex].
- Divide 117375 by 3:
- [tex]\(117375 \div 3 = 39125\)[/tex].
- Divide 39125 by 5:
- [tex]\(39125 \div 5 = 7825\)[/tex].
- Divide 7825 by 5:
- [tex]\(7825 \div 5 = 1565\)[/tex].
- Divide 1565 by 5:
- [tex]\(1565 \div 5 = 313\)[/tex].
- 313 is a prime number.
- Thus, the prime factorization of 234750 is [tex]\(2 \times 3 \times 5^3 \times 313\)[/tex].
I hope this step-by-step approach helps you understand how each number is broken down into its prime factors!
1. Prime Factorization of 252:
- Start by dividing 252 by the smallest prime, 2:
- [tex]\(252 \div 2 = 126\)[/tex].
- Divide 126 by 2:
- [tex]\(126 \div 2 = 63\)[/tex].
- 63 is not divisible by 2, so move to the next smallest prime, 3:
- [tex]\(63 \div 3 = 21\)[/tex].
- Divide 21 by 3:
- [tex]\(21 \div 3 = 7\)[/tex].
- 7 is a prime number, so we stop here.
- Thus, the prime factorization of 252 is [tex]\(2^2 \times 3^2 \times 7\)[/tex].
2. Prime Factorization of 360:
- Start with 360 and divide by 2:
- [tex]\(360 \div 2 = 180\)[/tex].
- Divide 180 by 2:
- [tex]\(180 \div 2 = 90\)[/tex].
- Divide 90 by 2:
- [tex]\(90 \div 2 = 45\)[/tex].
- 45 is not divisible by 2, move to 3:
- [tex]\(45 \div 3 = 15\)[/tex].
- Divide 15 by 3:
- [tex]\(15 \div 3 = 5\)[/tex].
- 5 is a prime number.
- So, the prime factorization of 360 is [tex]\(2^3 \times 3^2 \times 5\)[/tex].
3. Prime Factorization of 848:
- Divide 848 by 2:
- [tex]\(848 \div 2 = 424\)[/tex].
- Divide 424 by 2:
- [tex]\(424 \div 2 = 212\)[/tex].
- Divide 212 by 2:
- [tex]\(212 \div 2 = 106\)[/tex].
- Divide 106 by 2:
- [tex]\(106 \div 2 = 53\)[/tex].
- 53 is a prime number.
- Therefore, the prime factorization of 848 is [tex]\(2^4 \times 53\)[/tex].
4. Prime Factorization of 2890:
- Divide 2890 by 2:
- [tex]\(2890 \div 2 = 1445\)[/tex].
- 1445 is not divisible by 2, check with 5:
- [tex]\(1445 \div 5 = 289\)[/tex].
- 289 is not divisible by 5, check with other primes (it's actually [tex]\(17^2\)[/tex]).
- Prime factorization of 2890 is [tex]\(2 \times 5 \times 17^2\)[/tex].
5. Prime Factorization of 28910:
- Divide 28910 by 2:
- [tex]\(28910 \div 2 = 14455\)[/tex].
- 14455 is divisible by 5:
- [tex]\(14455 \div 5 = 2891\)[/tex].
- 2891 is divisible by 7:
- [tex]\(2891 \div 7 = 413\)[/tex].
- Finally, divide 413 by the next prime:
- [tex]\(413 \div 59 = 7\)[/tex].
- 7 is a prime.
- Prime factorization of 28910 is [tex]\(2 \times 5 \times 7^2 \times 59\)[/tex].
6. Prime Factorization of 234750:
- Divide 234750 by 2:
- [tex]\(234750 \div 2 = 117375\)[/tex].
- Divide 117375 by 3:
- [tex]\(117375 \div 3 = 39125\)[/tex].
- Divide 39125 by 5:
- [tex]\(39125 \div 5 = 7825\)[/tex].
- Divide 7825 by 5:
- [tex]\(7825 \div 5 = 1565\)[/tex].
- Divide 1565 by 5:
- [tex]\(1565 \div 5 = 313\)[/tex].
- 313 is a prime number.
- Thus, the prime factorization of 234750 is [tex]\(2 \times 3 \times 5^3 \times 313\)[/tex].
I hope this step-by-step approach helps you understand how each number is broken down into its prime factors!
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