If N = 10! then what is the minimum number which should be multiplied with N to make it a perfect square?
If N = 22 x 33 x 55, find the total number of odd and even factors of N.
M and N are two positive integers and each of them has 20 factors. If r is the total number of prime factors of M and s is the total number of prime factors of N, then what is the maximum value of rs?
Which of the following numbers is not a prime number?
If k is a positive integer, how many different prime factors does k have?
(1) k/60 is an integer
(2) k < 100
Steps 1 & 2: Understand Question and Draw Inferences
We need to find the prime factors of k
Step 3: Analyze Statement 1
k/60 is an integer
60 = 2^{2 }* 3 * 5
> 2, 3 and 5 are prime factors of k for sure
Not sufficient. We cannot say if 2, 3, 5 are the ONLY prime factors of k or not
Step 4: Analyze Statement 2
k < 100
Not sufficient. We do not know anything about k’s prime factors
Step 5: Analyze Both Statements Together (if needed)
Inference from statement 1: 2, 3 and 5 are prime factors of k
Inference from statement 2: k < 100
Inference from statement 1 and statement 2: 7 is the next prime after 5, so if k contains 7, k will become greater than 100
> 2, 3 and 5 are the only prime factors of k
> k has 3 prime factors
Statement 1 and Statement 2 together are sufficient to answer the question.
Answer: Option (C)
If x is a positive integer less than 100, for how many values of x is x/6 a prime number?
Step 1: Question statement and Inferences
Given: x is a positive integer such that 0 < x < 100
To find: The number of values for which x/6 is a prime number
Let x/6 = P, where P is a prime number
> x = 6P
Now we are given:
0 < x < 100
> 0 < 6P < 100
> 0 < P < 16.67
So, we need to find the number of prime numbers between 0 and 16.67
Step 2: Finding required values
The prime numbers between 0 and 16.67 are:
2, 3, 5, 7, 11, 13
Step 3: Calculating the final answer
The number of prime numbers between 0 and 16.67 is 6
Therefore, the correct answer choice is B
Answer: Option (B)
How many odd factors does the number 2100 have ?
Step 1: Question statement and Inferences
We are given the number 2100. We have to find the total number of odd factors of this number.
Step 2: Finding required values
We know that to find the total number of factors, first we have to break down the number into its prime factors.
2100 = 2 * 2 * 3 * 5 * 5 * 7
The next step is to write the prime factors in exponential form.
2100 = 2^{2} * 3^{1 }* 5^{2 }* 7^{1}
Now, since we have to find the total number of odd factors, we cannot include 2 or any higher power of 2 while calculating the number of factors.
Only the powers of 3, 5 and 7 can be considered to determine the number of odd factors.
So, total number of odd factors = (Power of 3 + 1) * (Power of 5 + 1) * (Power of 7 + 1)
= (1+1)(2+1)(1+1)
=2*3*2
=12
Step 3: Calculating the final answer
So, the number of odd factors of 2100 = 12
Answer: Option (E)
If n is a positive integer, which of the following statements are correct?
Step 1: Question statement and Inferences
Let us list the prime numbers less than 50:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 . . .
Step 2: Finding required values
Let’s now evaluate the two forms given and then evaluate which of these is able to describe the list above.
Option A: The form 4n + 1 and 4n + 3
As we see, all the prime numbers greater than 3 can be represented in the form 4n+1 or 4n+3.
Therefore, Statement I is correct.
However, we also see in the table that not every number of the form 4n + 1 and 4n + 3 is a prime number.
Therefore, Statement II is not correct.
Option B: The form 6n – 1 and 6n + 1
As we see, all the prime numbers greater than 3 can be represented in the form 6n+1 or 6n1.
Therefore, Statement III is correct.
Step 3: Calculating the final answer
Looking at the answer choices, we see that Option E is correct.
What is the greatest prime factor of 12^{3} – 96?
Step 1: Question statement and Inferences
We are required to find the greatest prime factor of 12^{3} – 96
Hence, we need to break the expression into its prime factors
Step 2: Finding required values
Given: 12^{3} – 96
= (2^{2}*3) ^{3} – 2^{5} * 3
= 2^{6}*3^{3} – 2^{5} * 3
= 2^{5} * 3*(2*3^{2}1)
= 2^{5} * 3*(2* 9 1)
= 2^{5} * 3*(18 1)
= 2^{5} *3 *17
The prime factors of (12^{3} – 96) are 2, 3 and 17
Step 3: Calculating the final answer
Among the prime factors 2, 3 and 17
17 is the greatest in value
Hence, 17 is the greatest prime factor of 12^{3} – 96
Answer: Option (D)
If A and B are positive integers, does A have more prime factors than B?
(1) B is the square root of A
(2) A ÷ B is an integer
Steps 1 & 2: Understand Question and Draw Inferences
We need to find if A has more prime factors than B
Step 3: Analyze Statement 1
B is the square root of A
^{ }A = B^{2}
> A and B will have same prime factors
(If X and n are positive integers, X^{n} has the same prime factors as X)
Sufficient.
Step 4: Analyze Statement 2
A ÷ B is an integer
> A may or may not have more prime factors than B
Not sufficient.
Step 5: Analyze Both Statements Together (if needed)
We get a unique answer in step 3, so this step is not required
Answer: Option (A)
If the number 13 completely divides x, and x = a^{2} * b, where a and b are distinct prime numbers, which of these numbers must be divisible by 169?
Step 1: Question statement and Inferences
We are given that the number x is a multiple of 13.
This means, x = 13 * k (k is an integer) …….. (1)
Also,
x = a^{2} * b, where a and b are prime numbers. ………… (2)
Now, since x has only two distinct prime factors a and b, and x is a multiple of 13, one of the numbers a and b is 13.
So, either a = 13 or b =13.
We have to find the number from the given options that is definitely a multiple of 169.
169 = 13^{2}
Step 2: Finding required values
Now, if a = 13, we need the term a^{2} in the number to make it a multiple of 169.
And, if b = 13, we need the term b^{2} in the number to make it a multiple of 169.
We do not know which number out of a and b is equal to 13.
So, we can only be sure that a given number is divisible by 13^{2} if the powers of both a and b are 2 or higher in that number. In that case, whether a = 13 or b = 13, the number will be divisible by 13^{2} for sure.
There is only one such number in the options: a^{2} b^{2}.
Step 3: Calculating the final answer
So, the number that is definitely a multiple of 169 is a^{2} b^{2}.
Answer: Option (D)
Is the number of distinct prime factors of the positive integer X more than 4?
(1) X is a multiple of 42.
(2) X is a multiple of 98.
Steps 1 & 2: Understand Question and Draw Inferences
We are given that x is a positive integer. We have to find whether X has more than 4 distinct prime factors.
Since we are not given any other information, let’s move on to the analysis of the statements.
Step 3: Analyze Statement 1
Statement 1 says: X is a multiple of 42.
Per this statement, x can be represented in the following format:
X = 2^{p} * 3^{q} * 7^{r} *k_{1} , where k_{1}, p, q, and r are positive integers.
So, we know that 2, 3, and 7 are the factors of X. However, we don’t know whether there are any more prime factors of X or not. So, we can’t say if the distinct prime factors of X are more than 4.
Hence, statement I is not sufficient to answer the question: Is the number of distinct prime factors of the positive integer X more than 4?
Step 4: Analyze Statement 2
Statement 2 says: X is a multiple of 98.
Per this statement, X can be represented in the following format:
X = 2^{a} * 7^{b} *k_{2}, where k_{2}, a and b are positive integers, and b is greater than or equal to 2.
So, we know that 2 and 7 are the factors of X. However, we don’t know whether there are any more prime factors of X or not. So, we can’t say if the distinct prime factors of X are more than 4.
Hence, statement II is not sufficient to answer the question: Is the number of distinct prime factors of the positive integer X more than 4?
Step 5: Analyze Both Statements Together (if needed)
From Statement (1):
X = 2^{p} * 3^{q} * 7^{r} *k_{1} , where k_{1}, p, q, and r are positive integers.
From Statement (2):
X = 2^{a} * 7^{b} *k_{2}, where k_{2}, a and b are positive integers, and b is greater than or equal to 2.
From the combination of both the above statements, we get to know that 2, 3, and 7 are the factors of X. However, it is possible that there are more prime factors about which no information is provided in the question statement.
So, we can’t be sure whether X has more than 4 prime factors or not.
So, statement (1) and (2) combined are not sufficient to answer the question: Is the number of distinct prime factors of the positive integer X more than 4?
Answer: Option (E)
What is the total number of distinct prime factors of 28980?
Step 1: Question statement and Inferences
We are given the number 28980. We have to find the total number of prime factors of this number.
Step 2: Finding required values
We know that to find the total number of prime factors, first we have to break down the number into its prime factors.
28980 = 2 * 2 * 3 *3* 5 * 7 * 23
The next step is to write the prime factors in exponential form.
28980 = 2^{2} * 3^{2} * 5^{1} * 7^{1} * 23^{1 }
Now, the distinct prime factors of the given number are 2, 3, 5, 7, and 23. So, there are 5 distinct prime factors of the number 28980.
Step 3: Calculating the final answer
So, the number of distinct prime factors of 28980 is 5
Answer: Option (D)
Find the number of factors of 180 that are in the form (4*k + 2), where k is a nonnegative integer?
Step 1: Question statement and Inferences
We are given the number 180. We have to find the number factors of 180 that are in the form (4*k + 2), where k is an integer.
Now, let’s analyze the numbers of the form 4*k + 2.
4*k + 2 = 2*(2*k + 1) = 2*odd number
So, basically the numbers of the form (4*k + 2) are odd numbers multiplied by 2.
Step 2: Finding required values
We know that to find the total number of factors, first we have to break down the number into its prime factors.
180 = 2 * 2 * 3 * 3 * 5
The next step is to write the prime factors in exponential form.
180 = 2^{2} * 3^{2} * 5^{1 }
Now, since we have to find the factors in the form of 2*(2*k + 1) i.e. 2*odd number, we can include only 2^{1} in the factors. We cannot include any other power of 2 in the factors.
[Note: If we included 2^{0} then the factor will become an odd number whereas if we include 2^{2} the number will not be in the form 2*(2*k + 1).]
So, total number of odd factors = (1) * (2 + 1) * (1 + 1) = 6
Step 3: Calculating the final answer
So, the number of factors of 180 in the form 2*(2*k + 1) = 6
(These factors are 2, 6, 10, 18, 30, and 90. Note that each of the numbers contains only one power of 2.)
Answer: Option (E)
If a, b are two distinct prime number than highest common factor of a, b is
Explanation: HCF of two prime numbers is 1.
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