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Find the Number of Factors of 2^8 × 7^5 × 14
The number of factors of 2^8 × 7^5 × 14 is 70.
To find the number of factors of 2^8 × 7^5 × 14, we’ll first express the number in its prime factorization form.
14 = 2 × 7
So, the prime factorization of 2^8 × 7^5 × 14 is:
2^8 × 7^5 × 2 × 7
Now, we rewrite this as:
2^(8+1) × 7^(5+1) = 2^9 × 7^6
Now, we can use the formula:
Total Number of Factors = (a+1) × (b+1)
Where a and b are the exponents of the prime factors 2 and 7 respectively.
So, plugging in the values:
Total Number of Factors = (9+1) × (6+1) = 10 × 7 = 70
Therefore, the number of factors of 2^8 × 7^5 × 14 is 70.
Factors of Square Numbers
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Even Powers of Prime Numbers: The prime factorization of a square number will always contain even powers of prime numbers. For example, the prime factorization of 16 is 2^4, where the exponent 4 is even.
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Symmetric Factors: Square numbers have a unique property where their factors are symmetric around the square root of the number. For example, the factors of 16 are 1, 2, 4, 8, and 16. Notice that they are symmetric around 4, which is the square root of 16.
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Pairs of Factors: Each factor of a square number comes in pairs. If n is a factor of a square number m, then m/n is also a factor. For example, the factors of 16 are 1, 2, 4, 8, and 16. We can pair them as (1, 16) and (2, 8), where each pair multiplies to 16.
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Square Roots: The square root of a square number will always be a whole number. This means that if n^2 = m, then n is a factor of m. For example, the square root of 16 is 4, which is a factor of 16.
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Range of Factors: The factors of a square number are always less than or equal to its square root. For example, the factors of 16 are 1, 2, 4, 8, and 16, all of which are less than or equal to the square root of 16, which is 4.
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Multiplicative Property: The factors of a square number can be obtained by multiplying different combinations of its prime factors. For example, the prime factors of 16 are 2^4, so the factors can be obtained by taking different combinations of these prime factors: 2^0, 2^1, 2^2, 2^3, and 2^4.
Understanding these properties can help in finding factors efficiently and understanding the structure of square numbers.
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Categories: Math
