Quant Boosters  Sibanand Pattnaik  Set 2

n^3 means exponent should be a multiple of 3
70= 10 * 7 i.e (9+1) * (6+1) both powers have to be multiple of 3

Q21) Let D be the number of divisors of a natural number N. Let D1 be the number of divisors of D, D2 the number of divisors of D1 and so on. What will be the final value of Dn?
a) 1
b) 2
c) 3
d) None of the above

You will finally end up with a prime number which will have 2 divisors. 2 has 2 divisors. Hence (b)

Q22) In how many ways can 3660 be written as the sum of 2 or more consecutive positive integers?
a) 2
b) 3
c) 4
d) More than 4

Direct formula for natural numbers = number of odd factors  1
So 3660 = 2^2 * 3 * 5 * 61
so odd factors = 8
Answer = 8  1 = 7
as in case of "integers" its 2 * (number of odd factors)  1
had it been asked "integers then ur answer would have been 16  1 = 15Else you can think this way :
3660 = 61 x 3 x4 x5
3660= a + (a+1) + (a+2)...+ (a+(n1)
Then 3660 = n x Average of the above series
If the number of terms in the series is odd, then the middle term of the series will be the average
So say the number of terms is 61, then the middle term will be 3x4x5= 60. So the series will be: (30+31+32...60+61+62...90)
If the number of terms is 3, the average will be 61x4x5= 1220. So the series is 1219+1220+1221.
Therefore if the number of terms is odd, its easy to construct a series like this
3660 has 7 odd factors other than 1

Q23) N has 6 factors and lies between 250 and 350. How many values of N are possible?
a) 12
b) 13
c) 14
d) None of the above

If a and b are prime numbers then N= a^5 or N= a^2 x b
N= a^5 :No values
N= a^2 x b
If a= 2, then b can take prime values of 67, 71, 73, 79, 83: 5 values
If a= 3, then b can take prime values of 29, 31, 37: 3 values
If a= 5, then b can take prime values of 11, 13: 2 values
Total values: 10

Q24) N^2 has 15 factors. M^2 has 9 factors. Which of the following cannot be the number of factors that MxN can have, if M and N are co prime to each other?
a) 24
b) 32
c) 20
d) 40

N= a^7 OR N= a^2xb^1
M= p^4 OR M= p^1xq^1
Where a,b, p and q are prime numbers
Therefore M x N=
i) a^7 x p^4 : 40 factors
ii) a^2 x b x p^4 : 30 factors
iii) a^7 x p x q : 32 factors
iv) a^2 x b x p x q= 24 factors

Q25) Sum of HCF and LCM of 2 numbers is 30. How many such pairs of numbers exist?
a) 6
b) 4
c) 7
d) 8

h(1+xy) = 30
h=1 xy=29 so 2^0=1
h=2 xy= 14 = 2*7 so 2^1= 2
h=3 xy=9 = 3^2 so 2^0= 1
h=5 xy=5 so 2^0=1
h=6 xy=4 so 2^0=1
h=10 xy=2 so 2^0=1
h=15 xy=1 so 1
> 1 + 2 + 5 * 1= 8

Q26) N is a natural number such that N/9 is a perfect square and N/4 is a perfect cube. Find how many factors does the smallest value of N have?
a) 18
b) 24
c) 14
d) 21

N = 2^2 * 3^6
factors = 21

Q27) How many ways can 1500 be expressed as product of 3 integers ?

1500 = 2^2 *3 * 5^3
ordered solutions = 4c2 * 3c2 * 5c2 = 180
integral : here we can have any 2 out 3 elements as negative ( for example , xyz = x * y * z)
so 3c2 = 3 ways so total 180 * 3 = 540 negative solutions
so finally 180 + 540 = 720 ordered integral solutions
total perfect squares in 1500 = 4
Total cubes = 1
So (a,a,b) cases = 3 and (a,a,a) cases = 1
now for integral (a,a,b), each of it can be (a,a,b) and (a,a,b)
finally 2 * 4 1 = 7 aab types and only 1 aaa type
unordered = [720  3 * 7 1]/6 + 8 = 124
so its 124 ways

Q28) How many 2 digit numbers are there such that the sum of their digits is a prime number?
a) 31
b) 33
c) 25
d) 27

Sum of the digits is prime means the sum of the digits can be
2: 11,20
3:12,21,30
5:41,14,23,32,50
7:61,16,25,52,34,43,70
11:92,29,83,38,74,47,65,56
13:94,49,85,58,76,67
17:98,89
There are 33 such numbers. Hence (b)

Q29) N has 12 factors. Which of the following cannot be the number of factors N^2?
a) 33
b) 35
c) 45
d) 54

N can be of any of the following form:
P^5x Q^1 therefore N^2 = P^10x Q^2; therefore N^2 has 33 factors
P^3x Q^2 therefore N^2 = P^6x Q^4; therefore N^2 has 35 factors
P^2x Q^1 x R^1 therefore N^2 = P^4x Q^2 x R^2; therefore N^2 has 45 factors
Therefore 54 is the answer

Q30) How many factors (x, y, z) of 1000 exist such that HCF (x,y,z) = 1 ?