Please explain on how to do each of these! *Let (x) be defined for all positive integer values of xas the product of all even factors of 4x. For example, (3)=12x6x4x2=576. What is the value of (5)? Someone says that it is f(5)= 20x10x2x4=1600 I DO NOT GET HOW THEY ARRIVED THAT ANSWER!
* 2.3 ^5 log(is raised at the same level as 5 meaning not above 5) base 2.3 ^1 The answer is 1 but I do not get how?
*If p and q are two different prime numbers greater than 2, and n=pq, how many positive factors, including 1 and n, does n have? Do not get how the answer is 4.
1>
even factors of 4*5 =20
so the factors are 1, 2 ,4 ,5 ,10 ,20
among them 2,4,10,20 are even.
3>
here 4 is the minimum no of +ve factors.
n will have the factors,
=1*n
=p*q
including 1 and n, n have minimum 4 factors.
it could be more.
***for Q2 I have not understood.
1. Let (x) be defined as the product of all even factors of 4x. In order to find the value of (5), we need to determine the even factors of 4x where x = 5.
First, let's find the factors of 4x:
Factors of 4x = 1, 2, 4, x, 2x, 4x (since 4x is an even number)
Now, substitute x = 5:
Factors of 4(5) = 1, 2, 4, 5, 2(5), 4(5) = 1, 2, 4, 5, 10, 20
To calculate (5), we need to multiply all the even factors together:
(5) = 2 * 4 * 10 * 20 = 1600
So, the correct calculation for (5) is indeed f(5) = 1600.
2. To solve 2.3 ^5 log(base 2.3) ^1, we need to understand the properties of logarithms.
The base of a logarithm indicates the number that is raised to a certain power in order to obtain the given value. In this case, the base of the logarithm is 2.3.
According to logarithm rules, when we have log(base a) of a (where a is a positive number), the value of the logarithm will be equal to 1.
Applying this to the given expression:
2.3 ^5 log(base 2.3) ^1 = 2.3 ^5 * 1 = 2.3 ^5
Therefore, the answer is 2.3 ^5 = 1.
3. Given that p and q are two different prime numbers greater than 2, and n = pq, we need to find the number of positive factors of n.
When we multiply two different prime numbers, the resulting number will have four factors, which are 1, p, q, and n.
So, the positive factors of n = pq are 1, p, q, and n.
Hence, the answer is 4 factors.
1. Let's break down the first question step by step. We need to find the value of (5), which is defined as the product of all even factors of 4x, where x = 5 in this case.
First, we determine the even factors of 4x. In this case, 4x = 4 * 5 = 20. The even factors of 20 are 2, 4, 10, and 20.
Secondly, we need to determine the product of these even factors. So, we multiply all these factors together: 2 * 4 * 10 * 20 = 1600.
Therefore, the correct answer is f(5) = 1600, not 20x10x2x4 = 1600. It seems that the person who provided the incorrect answer made a mistake in their calculations.
2. For the second question, we are asked to calculate the expression 2.3^5 log base 2.3^1.
First, let's simplify the expression: 2.3^5 log base 2.3^1 = 2.3^5 log base 2.3.
Now, since the base of the logarithm is the same as the base of the exponent, we can use the property of logarithms that states log base a (a^b) = b.
So, the expression simplifies to: 2.3^5 * 1 = 2.3^5 = 1.
Therefore, the answer is 1.
3. In the third question, we are given two different prime numbers greater than 2, denoted as p and q. We are asked to find the number of positive factors of n, where n = p * q.
Firstly, let's understand what it means to be a factor of a number. A factor of a number is a whole number that divides the given number without any remainder.
In this case, since p and q are both prime numbers, the only positive factors of n = p * q are 1, p, q, and n itself.
Therefore, the answer is 4, including 1 and n.