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31 May 2022, 13:39
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If N lies between 10 and 30 and the total number of factors of N is 4, how many such N are possible?
A. 4
B. 5
C. 6
D. 7
E. 8
A. 4
B. 5
C. 6
D. 7
E. 8
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31 May 2022, 14:00
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since N is between 10<N<30
and n has only 4 factor that means n has 2 prime factors apart from 1 and N itself
so we have to find pairs of prime factors having product between 10 to 30
2*7 , 2*11, 2*13
3*5, 3*7
total 5
Also there is perfect cube in the range
27 = 1 * 3 * 3 * 3
having factors 1,3,9,27
total =1
so total numbers 5+ 1 =6
and n has only 4 factor that means n has 2 prime factors apart from 1 and N itself
so we have to find pairs of prime factors having product between 10 to 30
2*7 , 2*11, 2*13
3*5, 3*7
total 5
Also there is perfect cube in the range
27 = 1 * 3 * 3 * 3
having factors 1,3,9,27
total =1
so total numbers 5+ 1 =6
01 Jun 2022, 07:48
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N lies between 10 and 30. The total number of factors of N is 4, therefore, N is a composite number.
For a composite number in the form of K = \((P_1)^x\) * \((P_2)^y\) * \((P_3)^z\) * …., the number of factors = (x + 1) (y + 1) (z + 1)…….
Since N is composite number and the number of factors = 4,
(x + 1) (y + 1) (z + 1)… = 4
4 is a composite number; hence it can be broken down in more than one ways.
Case 1:
(x + 1) = 4 and (y + 1) = 1
Therefore, x = 3 and y = 0.
This means, the number is of the form \((P_1)^3\); which means that N is the cube of a prime number.
This is satisfied by only one number viz., 27.
Case 2:
(x + 1) = 2 and (y + 1) = 2
Therefore, x = 1 and y = 1.
This means, the number is of the form \((P_1)^1\) * \((P_2)^1\); which means that N is the product of two distinct primes.
This is satisfied by the numbers 14, 15, 21, 22 and 26.
Therefore, total number of values possible for N = 1 + 5 = 6.
The correct answer option is C.
For a composite number in the form of K = \((P_1)^x\) * \((P_2)^y\) * \((P_3)^z\) * …., the number of factors = (x + 1) (y + 1) (z + 1)…….
Since N is composite number and the number of factors = 4,
(x + 1) (y + 1) (z + 1)… = 4
4 is a composite number; hence it can be broken down in more than one ways.
Case 1:
(x + 1) = 4 and (y + 1) = 1
Therefore, x = 3 and y = 0.
This means, the number is of the form \((P_1)^3\); which means that N is the cube of a prime number.
This is satisfied by only one number viz., 27.
Case 2:
(x + 1) = 2 and (y + 1) = 2
Therefore, x = 1 and y = 1.
This means, the number is of the form \((P_1)^1\) * \((P_2)^1\); which means that N is the product of two distinct primes.
This is satisfied by the numbers 14, 15, 21, 22 and 26.
Therefore, total number of values possible for N = 1 + 5 = 6.
The correct answer option is C.
