the cube of the first no exceed their product by 2,the cube of the second no is smaller than their product by 3,and the cube of the third no exceeds their product by 3

A)3^(1/3) B)9^(1/3) C)2 D)any of these

For how many integers x^{4} + x^{3} + x^{2} + x + 1 is a perfect square?

For how many integers, n, the expression n^{2} + n + 4 is a perfect square?

for only 1 integer ?

4 integers

-3 , -1 , 0 and 4.

Sir please share the approach

Q.How many natural numbers between 1 and 100 have exactly 4 factors?

For exactly 4 factors . Result comes from following 2 forms

2*2---> corresponds to the number of form (a*b) where a,b are primes

and 4 ---> corresponds to the number of form (a^3)

Now let's take each form one by one

a*b form

2*3, 2*3, 2*5 ...........2*47 (14 numbers) (14 primes from 3 to 47)

then 3*5, 3*7,3*11........ 3*31 (9 numbers) (9 primes from 5 to 31)

then 5*7, 5*11.......5*19 (5 numbers)

then 7*11, 7*13 (2 numbers)

total number of this form = 30 numbers

next is a^3 form

only 3 numbers are there 2^3, 3^3, 4^3

so total 33 numbers have exactly 4 factors

10000! =(100!)^k ×p where k and p are integers. What can be the maximum value of K?

On planet Mars the people use a certain number system to the base 'n' (n>2) , kyes a resident of the planet, one day received thrice of his daily wage because the digits of this wage, which was a 2 digit number, were reversed.

Q1. What is the least possible value of n?

Q2. What is the second least possible value of n?

Q3. If n is least possible value then what is the wages that kyes receives on a normal day (in base n)?

Q4. If the base system that is being used on the mars is the second least possible value of N then what is the sum of all the value of wages that kyes can receive for his work (in base n)?

product is an integer, hence x and y must take same value.

now first factorize 2016

2016= 63*32

so we can take the above expression as

32k(8^2-p)

now p could be such that inside we are left with a factor of 63.

Hence when p =43 , we are left with 21 the k will become 3

and p=61, then we are left with 3, then k will become 21.

These are the only 2 values hence sum of values of p is 104

median is the middle position

Now when increase of 10 changes the avg by 2. Hence there are 5 terms.

10, m-4, m , 32, 32

m is in the middle and m-4 immediate left. because in the second case when m-8 is taken instead m, then the median is m-4.

now sum of the terms should be 22*5= 110

hence m+m-4 = 36

hence m = 20