Find the total number of integer `n` such that `2lt=nlt=2000` and H.C.F. of `n` and 36 is 1.

The sum of all natural numbers 'n' such that 100 lt n lt 200 and HCF (91, n) gt 1 is

Find the least positive integer n such that 1 + 6 +6^2 + … + 6^n gt 5000

n_1a n dn_2 are four-digit numbers, find the total number of ways of forming n_1a n dn_2 so that n_2 can be subtracted from n_1 without borrowing at any stage.

Find the least positive integer n such that 1 + 6 + 6^(2) + …. + 6^(n) gt 5000 .

If n_1 and n_2 are five-digit numbers, find the total number of ways of forming n_1 and n_2 so that these numbers can be added without carrying at any stage.

Find the total number of n -digit number (n >1) having property that no two consecutive digits are same.

Two players P_1a n dP_2 play a series of 2n games. Each game can result in either a win or a loss for P_1dot the total number of ways in which P_1 can win the series of these games is equal to a. 1/2(2^(2n)-^ "^(2n)C_n) b. 1/2(2^(2n)-2xx^"^(2n)C_n) c. 1/2(2^n-^"^(2n)C_n) d. 1/2(2^n-2xx^"^(2n)C_n)

Find the total number of parallel tangents of f_1(x)=x^2-x+1a n df_2(x)=x^3-x^2-2x+1.