Answer :
Let's break down each number step-by-step to express it as a product of its prime factors:
### a. 150
1. Divide by the smallest prime number, 2:
- [tex]\(150 \div 2 = 75\)[/tex]
2. Next, divide 75 by the next smallest prime number, 3:
- [tex]\(75 \div 3 = 25\)[/tex]
3. Finally, divide 25 by the prime number, 5 (since 25 is [tex]\(5 \times 5\)[/tex]):
- [tex]\(25 \div 5 = 5\)[/tex]
- [tex]\(5 \div 5 = 1\)[/tex]
So, [tex]\(150 = 2 \times 3 \times 5 \times 5 = 2 \times 3 \times 5^2\)[/tex].
### b. 725
1. Divide by the smallest prime number, 5:
- [tex]\(725 \div 5 = 145\)[/tex]
2. Next, divide 145 by the smallest prime number, 5:
- [tex]\(145 \div 5 = 29\)[/tex]
3. 29 is a prime number, so it cannot be divided further.
So, [tex]\(725 = 5 \times 5 \times 29 = 5^2 \times 29\)[/tex].
### c. 230
1. Divide by the smallest prime number, 2:
- [tex]\(230 \div 2 = 115\)[/tex]
2. Next, divide 115 by the next smallest prime number, 5:
- [tex]\(115 \div 5 = 23\)[/tex]
3. 23 is a prime number, so it cannot be divided further.
So, [tex]\(230 = 2 \times 5 \times 23\)[/tex].
### d. 732
1. Divide by the smallest prime number, 2:
- [tex]\(732 \div 2 = 366\)[/tex]
2. Divide 366 by 2 again:
- [tex]\(366 \div 2 = 183\)[/tex]
3. Next, divide 183 by the next smallest prime number, 3:
- [tex]\(183 \div 3 = 61\)[/tex]
4. 61 is a prime number, so it cannot be divided further.
So, [tex]\(732 = 2 \times 2 \times 3 \times 61 = 2^2 \times 3 \times 61\)[/tex].
### e. 256
1. Since 256 is a power of 2, divide by 2 continuously:
- [tex]\(256 \div 2 = 128\)[/tex]
- [tex]\(128 \div 2 = 64\)[/tex]
- [tex]\(64 \div 2 = 32\)[/tex]
- [tex]\(32 \div 2 = 16\)[/tex]
- [tex]\(16 \div 2 = 8\)[/tex]
- [tex]\(8 \div 2 = 4\)[/tex]
- [tex]\(4 \div 2 = 2\)[/tex]
- [tex]\(2 \div 2 = 1\)[/tex]
So, [tex]\(256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^8\)[/tex].
### f. 67
1. 67 is a prime number, so it cannot be divided further.
So, [tex]\(67 = 67\)[/tex].
#### Summary
Here are the numbers expressed as products of their prime factors:
a. [tex]\(150 = 2^1 \times 3^1 \times 5^2\)[/tex]
b. [tex]\(725 = 5^2 \times 29\)[/tex]
c. [tex]\(230 = 2^1 \times 5^1 \times 23^1\)[/tex]
d. [tex]\(732 = 2^2 \times 3^1 \times 61^1\)[/tex]
e. [tex]\(256 = 2^8\)[/tex]
f. [tex]\(67 = 67\)[/tex]
### a. 150
1. Divide by the smallest prime number, 2:
- [tex]\(150 \div 2 = 75\)[/tex]
2. Next, divide 75 by the next smallest prime number, 3:
- [tex]\(75 \div 3 = 25\)[/tex]
3. Finally, divide 25 by the prime number, 5 (since 25 is [tex]\(5 \times 5\)[/tex]):
- [tex]\(25 \div 5 = 5\)[/tex]
- [tex]\(5 \div 5 = 1\)[/tex]
So, [tex]\(150 = 2 \times 3 \times 5 \times 5 = 2 \times 3 \times 5^2\)[/tex].
### b. 725
1. Divide by the smallest prime number, 5:
- [tex]\(725 \div 5 = 145\)[/tex]
2. Next, divide 145 by the smallest prime number, 5:
- [tex]\(145 \div 5 = 29\)[/tex]
3. 29 is a prime number, so it cannot be divided further.
So, [tex]\(725 = 5 \times 5 \times 29 = 5^2 \times 29\)[/tex].
### c. 230
1. Divide by the smallest prime number, 2:
- [tex]\(230 \div 2 = 115\)[/tex]
2. Next, divide 115 by the next smallest prime number, 5:
- [tex]\(115 \div 5 = 23\)[/tex]
3. 23 is a prime number, so it cannot be divided further.
So, [tex]\(230 = 2 \times 5 \times 23\)[/tex].
### d. 732
1. Divide by the smallest prime number, 2:
- [tex]\(732 \div 2 = 366\)[/tex]
2. Divide 366 by 2 again:
- [tex]\(366 \div 2 = 183\)[/tex]
3. Next, divide 183 by the next smallest prime number, 3:
- [tex]\(183 \div 3 = 61\)[/tex]
4. 61 is a prime number, so it cannot be divided further.
So, [tex]\(732 = 2 \times 2 \times 3 \times 61 = 2^2 \times 3 \times 61\)[/tex].
### e. 256
1. Since 256 is a power of 2, divide by 2 continuously:
- [tex]\(256 \div 2 = 128\)[/tex]
- [tex]\(128 \div 2 = 64\)[/tex]
- [tex]\(64 \div 2 = 32\)[/tex]
- [tex]\(32 \div 2 = 16\)[/tex]
- [tex]\(16 \div 2 = 8\)[/tex]
- [tex]\(8 \div 2 = 4\)[/tex]
- [tex]\(4 \div 2 = 2\)[/tex]
- [tex]\(2 \div 2 = 1\)[/tex]
So, [tex]\(256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^8\)[/tex].
### f. 67
1. 67 is a prime number, so it cannot be divided further.
So, [tex]\(67 = 67\)[/tex].
#### Summary
Here are the numbers expressed as products of their prime factors:
a. [tex]\(150 = 2^1 \times 3^1 \times 5^2\)[/tex]
b. [tex]\(725 = 5^2 \times 29\)[/tex]
c. [tex]\(230 = 2^1 \times 5^1 \times 23^1\)[/tex]
d. [tex]\(732 = 2^2 \times 3^1 \times 61^1\)[/tex]
e. [tex]\(256 = 2^8\)[/tex]
f. [tex]\(67 = 67\)[/tex]
