To solve this problem, we need to understand the properties of prime factors and the number of factors a number can have.

1. The prime factorization of a number is the expression of that number as a product of primes. For example, the prime factorization of 18 is 2*3*3.

2. The number of factors a number has can be determined by expressing the number as a product of powers of prime numbers (its prime factorization), then adding 1 to each of the exponents, and multiplying them together.

Given that the number has 24 factors, it can be expressed in the form of p^a * q^b * r^c * s^d, where p, q, r, s are prime numbers and a, b, c, d are their respective powers. The number of factors is then (a+1)(b+1)(c+1)(d+1) = 24.

Given that the sum of a particular set of 3 prime factors is 25 and the product of all 4 prime factors is 1365, we can list down the prime factors of 1365 which are 3, 5, 7, and 13.

The sum of 3, 5, and 7 is 15, and the sum of 3, 5, and 13 is 21, and the sum of 3, 7, and 13 is 23, and the sum of 5, 7, and 13 is 25. So, the set of 3 prime factors that sum up to 25 are 5, 7, and 13.

Now, we need to find the powers of these prime factors such that the number of factors is 24. The only way to get 24 by multiplying 4 numbers where each number is greater than or equal to 2 (since the power of a prime factor is at least 1, and we add 1 to the power to get the number of factors) is 2*2*2*3.

So, we can assign the powers 1, 1, 1, and 2 to the prime factors. The prime factor with the power of 2 should be 3 (the prime factor that is not in the set that sums up to 25), because if it was any of the other prime factors, the number would have more than 24 factors.

Therefore, the number that Bharathi has in mind is 3^2 * 5^1 * 7^1 * 13^1 = 9 * 5 * 7 * 13 = 4095.

Question

To solve this problem, we need to understand the properties of prime factors and the number of factors a number can have.

1. The prime factorization of a number is the expression of that number as a product of primes. For example, the prime factorization of 18 is 2*3*3.

2. The number of factors a number has can be determined by expressing the number as a product of powers of prime numbers (its prime factorization), then adding 1 to each of the exponents, and multiplying them together.

Given that the number has 24 factors, it can be expressed in the form of p^a * q^b * r^c * s^d, where p, q, r, s are prime numbers and a, b, c, d are their respective powers. The number of factors is then (a+1)(b+1)(c+1)(d+1) = 24.

Given that the sum of a particular set of 3 prime factors is 25 and the product of all 4 prime factors is 1365, we can list down the prime factors of 1365 which are 3, 5, 7, and 13.

The sum of 3, 5, and 7 is 15, and the sum of 3, 5, and 13 is 21, and the sum of 3, 7, and 13 is 23, and the sum of 5, 7, and 13 is 25. So, the set of 3 prime factors that sum up to 25 are 5, 7, and 13.

Now, we need to find the powers of these prime factors such that the number of factors is 24. The only way to get 24 by multiplying 4 numbers where each number is greater than or equal to 2 (since the power of a prime factor is at least 1, and we add 1 to the power to get the number of factors) is 2*2*2*3.

So, we can assign the powers 1, 1, 1, and 2 to the prime factors. The prime factor with the power of 2 should be 3 (the prime factor that is not in the set that sums up to 25), because if it was any of the other prime factors, the number would have more than 24 factors.

Therefore, the number that Bharathi has in mind is 3^2 * 5^1 * 7^1 * 13^1 = 9 * 5 * 7 * 13 = 4095.

Knowee AI · Accepted Answer