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Find the number of factors which are perfect square or perfect cube but not both in two ki power 7 into 3 ki power 8 into 5ki power9and 7 ki power 10?
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Find the number of factors which are perfect square or perfect cube bu...
To find the number of factors that are perfect squares or perfect cubes but not both in the given expression, we need to understand the concept of perfect squares and perfect cubes and how to find factors.
1. Understanding Perfect Squares and Perfect Cubes:
- A perfect square is a number that can be expressed as the square of an integer. For example, 4, 9, 16, etc.
- A perfect cube is a number that can be expressed as the cube of an integer. For example, 8, 27, 64, etc.
2. Understanding Factors:
- Factors of a number are the numbers that divide the given number without leaving any remainder.
- For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
3. Finding Factors of the Given Expression:
- The given expression is 2^7 * 3^8 * 5^9 * 7^10.
- To find the factors, we need to consider the powers of each prime number separately.
- The powers of the prime numbers are 7, 8, 9, and 10 for 2, 3, 5, and 7 respectively.
4. Factors of 2:
- The factors of 2 can be calculated by considering the powers of 2 from 0 to 7.
- The factors are 2^0, 2^1, 2^2, ..., 2^7.
- So, there are 8 factors of 2.
5. Factors of 3:
- The factors of 3 can be calculated by considering the powers of 3 from 0 to 8.
- The factors are 3^0, 3^1, 3^2, ..., 3^8.
- So, there are 9 factors of 3.
6. Factors of 5:
- The factors of 5 can be calculated by considering the powers of 5 from 0 to 9.
- The factors are 5^0, 5^1, 5^2, ..., 5^9.
- So, there are 10 factors of 5.
7. Factors of 7:
- The factors of 7 can be calculated by considering the powers of 7 from 0 to 10.
- The factors are 7^0, 7^1, 7^2, ..., 7^10.
- So, there are 11 factors of 7.
8. Finding Perfect Squares and Perfect Cubes:
- To find the perfect squares, we need to consider the factors with even powers.
- To find the perfect cubes, we need to consider the factors with powers divisible by 3.
9. Counting Factors that are Perfect Squares or Perfect Cubes but not both:
- The total number of factors in the given expression is (7+1) * (8+1) * (9+1) * (10+1) = 8 * 9 * 10 * 11 = 7,920.
- To find the factors that are perfect squares or perfect cubes but not both, we need to count the factors with even powers and factors with powers divisible by 3.
- For each prime number, we have even powers (0, 2, 4, 6), and powers divisible by 3
1. Understanding Perfect Squares and Perfect Cubes:
- A perfect square is a number that can be expressed as the square of an integer. For example, 4, 9, 16, etc.
- A perfect cube is a number that can be expressed as the cube of an integer. For example, 8, 27, 64, etc.
2. Understanding Factors:
- Factors of a number are the numbers that divide the given number without leaving any remainder.
- For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
3. Finding Factors of the Given Expression:
- The given expression is 2^7 * 3^8 * 5^9 * 7^10.
- To find the factors, we need to consider the powers of each prime number separately.
- The powers of the prime numbers are 7, 8, 9, and 10 for 2, 3, 5, and 7 respectively.
4. Factors of 2:
- The factors of 2 can be calculated by considering the powers of 2 from 0 to 7.
- The factors are 2^0, 2^1, 2^2, ..., 2^7.
- So, there are 8 factors of 2.
5. Factors of 3:
- The factors of 3 can be calculated by considering the powers of 3 from 0 to 8.
- The factors are 3^0, 3^1, 3^2, ..., 3^8.
- So, there are 9 factors of 3.
6. Factors of 5:
- The factors of 5 can be calculated by considering the powers of 5 from 0 to 9.
- The factors are 5^0, 5^1, 5^2, ..., 5^9.
- So, there are 10 factors of 5.
7. Factors of 7:
- The factors of 7 can be calculated by considering the powers of 7 from 0 to 10.
- The factors are 7^0, 7^1, 7^2, ..., 7^10.
- So, there are 11 factors of 7.
8. Finding Perfect Squares and Perfect Cubes:
- To find the perfect squares, we need to consider the factors with even powers.
- To find the perfect cubes, we need to consider the factors with powers divisible by 3.
9. Counting Factors that are Perfect Squares or Perfect Cubes but not both:
- The total number of factors in the given expression is (7+1) * (8+1) * (9+1) * (10+1) = 8 * 9 * 10 * 11 = 7,920.
- To find the factors that are perfect squares or perfect cubes but not both, we need to count the factors with even powers and factors with powers divisible by 3.
- For each prime number, we have even powers (0, 2, 4, 6), and powers divisible by 3
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Find the number of factors which are perfect square or perfect cube but not both in two ki power 7 into 3 ki power 8 into 5ki power9and 7 ki power 10?
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Find the number of factors which are perfect square or perfect cube but not both in two ki power 7 into 3 ki power 8 into 5ki power9and 7 ki power 10? for SSC CGL 2025 is part of SSC CGL preparation. The Question and answers have been prepared according to the SSC CGL exam syllabus. Information about Find the number of factors which are perfect square or perfect cube but not both in two ki power 7 into 3 ki power 8 into 5ki power9and 7 ki power 10? covers all topics & solutions for SSC CGL 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the number of factors which are perfect square or perfect cube but not both in two ki power 7 into 3 ki power 8 into 5ki power9and 7 ki power 10?.
Find the number of factors which are perfect square or perfect cube but not both in two ki power 7 into 3 ki power 8 into 5ki power9and 7 ki power 10? for SSC CGL 2025 is part of SSC CGL preparation. The Question and answers have been prepared according to the SSC CGL exam syllabus. Information about Find the number of factors which are perfect square or perfect cube but not both in two ki power 7 into 3 ki power 8 into 5ki power9and 7 ki power 10? covers all topics & solutions for SSC CGL 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the number of factors which are perfect square or perfect cube but not both in two ki power 7 into 3 ki power 8 into 5ki power9and 7 ki power 10?.
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