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What is the difference of exponents of prime factors in prime factorization of 1225?
- a)1
- b)2
- c)0
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
What is the difference of exponents of prime factors in prime factoriz...
Here 1225 = 52 × 72
∴ Required difference = 2 - 2 = 0
∴ Required difference = 2 - 2 = 0
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What is the difference of exponents of prime factors in prime factoriz...
To find the prime factorization of 1225, we need to determine the prime numbers that divide 1225 without leaving a remainder.
Prime factorization of 1225:
Step 1: Divide 1225 by the smallest prime number, which is 2. Since 1225 is not divisible by 2, we move to the next prime number.
Step 2: Divide 1225 by the next prime number, which is 3. Again, 1225 is not divisible by 3.
Step 3: Divide 1225 by the next prime number, which is 5. This time, 1225 is divisible by 5. 1225 ÷ 5 = 245.
Step 4: Now, we repeat the process with the quotient obtained from the previous step, which is 245.
Step 5: Divide 245 by the smallest prime number, which is 2. 245 is not divisible by 2.
Step 6: Divide 245 by the next prime number, which is 3. 245 is not divisible by 3.
Step 7: Divide 245 by the next prime number, which is 5. 245 is divisible by 5. 245 ÷ 5 = 49.
Step 8: Repeat the process with the quotient obtained from the previous step, which is 49.
Step 9: Divide 49 by the smallest prime number, which is 2. 49 is not divisible by 2.
Step 10: Divide 49 by the next prime number, which is 3. 49 is not divisible by 3.
Step 11: Divide 49 by the next prime number, which is 5. 49 is not divisible by 5.
Step 12: Divide 49 by the next prime number, which is 7. 49 is divisible by 7. 49 ÷ 7 = 7.
Step 13: Repeat the process with the quotient obtained from the previous step, which is 7.
Step 14: Divide 7 by the smallest prime number, which is 2. 7 is not divisible by 2.
Step 15: Divide 7 by the next prime number, which is 3. 7 is not divisible by 3.
Step 16: Divide 7 by the next prime number, which is 5. 7 is not divisible by 5.
Step 17: Divide 7 by the next prime number, which is 7. 7 is divisible by 7. 7 ÷ 7 = 1.
Therefore, the prime factorization of 1225 is 5 × 5 × 7 × 7.
Now, let's compare the exponents of the prime factors in the prime factorization.
- The exponent of 5 is 2.
- The exponent of 7 is 2.
Since the exponents of both prime factors are the same (2), the difference of the exponents is 0.
Hence, the correct answer is option C: 0.
Prime factorization of 1225:
Step 1: Divide 1225 by the smallest prime number, which is 2. Since 1225 is not divisible by 2, we move to the next prime number.
Step 2: Divide 1225 by the next prime number, which is 3. Again, 1225 is not divisible by 3.
Step 3: Divide 1225 by the next prime number, which is 5. This time, 1225 is divisible by 5. 1225 ÷ 5 = 245.
Step 4: Now, we repeat the process with the quotient obtained from the previous step, which is 245.
Step 5: Divide 245 by the smallest prime number, which is 2. 245 is not divisible by 2.
Step 6: Divide 245 by the next prime number, which is 3. 245 is not divisible by 3.
Step 7: Divide 245 by the next prime number, which is 5. 245 is divisible by 5. 245 ÷ 5 = 49.
Step 8: Repeat the process with the quotient obtained from the previous step, which is 49.
Step 9: Divide 49 by the smallest prime number, which is 2. 49 is not divisible by 2.
Step 10: Divide 49 by the next prime number, which is 3. 49 is not divisible by 3.
Step 11: Divide 49 by the next prime number, which is 5. 49 is not divisible by 5.
Step 12: Divide 49 by the next prime number, which is 7. 49 is divisible by 7. 49 ÷ 7 = 7.
Step 13: Repeat the process with the quotient obtained from the previous step, which is 7.
Step 14: Divide 7 by the smallest prime number, which is 2. 7 is not divisible by 2.
Step 15: Divide 7 by the next prime number, which is 3. 7 is not divisible by 3.
Step 16: Divide 7 by the next prime number, which is 5. 7 is not divisible by 5.
Step 17: Divide 7 by the next prime number, which is 7. 7 is divisible by 7. 7 ÷ 7 = 1.
Therefore, the prime factorization of 1225 is 5 × 5 × 7 × 7.
Now, let's compare the exponents of the prime factors in the prime factorization.
- The exponent of 5 is 2.
- The exponent of 7 is 2.
Since the exponents of both prime factors are the same (2), the difference of the exponents is 0.
Hence, the correct answer is option C: 0.
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