High School

Find the following by prime factorization:

a) 900
b) 3234
c) 1323
d) 8400
e) 960
f) 999
g) 2304
h) 3125

Answer :

Prime factorization is the process of expressing a number as the product of its prime factors. This is useful for simplifying fractions, finding least common multiples, and understanding the properties of numbers. Let's go through the prime factorization process for each number provided:

[a] 900:

  1. Start by dividing 900 by 2 (since it is even), resulting in 450.
  2. Divide 450 by 2 again, resulting in 225.
  3. 225 is not even, so divide by 3, resulting in 75.
  4. Divide 75 by 3 again, resulting in 25.
  5. Finally, divide by 5 (since 25 is a multiple of 5), giving us 5, and then another 5.
  6. Thus, the prime factorization of 900 is:
    [tex]2^2 \times 3^2 \times 5^2[/tex]

[b] 3234:

  1. Divide 3234 by 2, resulting in 1617.
  2. 1617 is not even, but it is divisible by 3, resulting in 539.
  3. Since 539 is not divisible by 3, check for divisibility by 7, resulting in 77.
  4. Finally, 77 can be divided by 7 to get 11.
  5. Thus, the prime factorization of 3234 is:
    [tex]2 \times 3 \times 7^2 \times 11[/tex]

[c] 1323:

  1. 1323 is divisible by 3, giving 441.
  2. 441 is divisible by 3 again, resulting in 147.
  3. 147 is also divisible by 3, leading to 49.
  4. 49 can be divided by 7 twice (as it is a square), resulting in 7.
  5. Therefore, the prime factorization of 1323 is:
    [tex]3^3 \times 7^2[/tex]

[d] 8400:

  1. Divide 8400 by 2 repeatedly: 4200, 2100, 1050, 525.
  2. 525 is divisible by 3, giving 175.
  3. 175 is divisible by 5, resulting in 35, which can be divided by 5 again to give 7.
  4. Thus, the prime factorization of 8400 is:
    [tex]2^4 \times 3 \times 5^2 \times 7[/tex]

[e] 960:

  1. Continuously divide 960 by 2 to obtain 480, 240, 120, 60, 30, and finally 15.
  2. 15 is divisible by 3, giving 5.
  3. Consequently, the prime factorization of 960 is:
    [tex]2^6 \times 3 \times 5[/tex]

[f] 999:

  1. Divide 999 by 3 to get 333, and divide again by 3 to get 111.
  2. 111 is divisible by 3, resulting in 37.
  3. Since 37 is a prime number, the prime factorization of 999 is:
    [tex]3^3 \times 37[/tex]

[g] 2304:

  1. Keep dividing 2304 by 2: 1152, 576, 288, 144, 72, 36, 18, 9.
  2. Then, divide 9 by 3 three times.
  3. Therefore, the prime factorization of 2304 is:
    [tex]2^8 \times 3^2[/tex]

[h] 3125:

  1. Notice that 3125 is not divisible by 2 or 3. However, it is divisible by 5 repeatedly: 625, 125, 25, and 5.
  2. With four 5s, the prime factorization is:
    [tex]5^5[/tex]

This methodical approach ensures you correctly determine the prime factors of each number.

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