If `x_i=a_i b_i c+i,i=1,,23` are three-digit positive integer such that each `x_1` is a multiple of 19, then for some integers `n ,` prove that `|a_1a_2a_3b_1b_2b_3c_1c_2c_3|` is divisible by 19.

" if " x_(i) =a_(i) b_(i) C_(i), i= 1,2,3 are three- digit positive integer such that each x_(i) is a mulptiple of 19 then prove that det {{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}} is divisible by 19.

The least positive integer n such that ((2i)/(1+i))^(n) is a positive integer is 16b.8c.4d.2

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat ka n d vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec aa n d vec b . If the angle between aa n db is pi/6, then prove that |a_1a_2a_3b_1b_2b_3c_1c_2c_3|=1/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2)

The least positive integer n such that ((2i)^n)/((1-i)^(n-2)),i=sqrt(-1) , is a positive integer,is ______

Let vec a=a_1 hat i+a_2 hat j+a_2 hat k , vec b=b_1 hat i+a_2 hat j+b_2 hat k ,a n d vec c=c_1 hat i+c_2 hat j+c_2 hat k , be three non-zero vectors such that vec c is a unit vector perpendicular to both vectors vec aa n d vec b . If the angle between aa n db is pi//6, then |a_1a_2a_3b_1b_2b_3c_1c_2c_3|^2 is equal to 0 1 1/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2) 3/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2)(c1 2+c2 2+c3 2)

If a, b and c are positive integers such that (1)/(a+(1)/(b+(1)/(c+(1)/(2))))=(16)/(23) then what is the mean of a,b and c?

If 97/19 = a+ 1/(b+1/c) where a, b and c are positive integers, then what is the sum of a, b and c?

The smallest positive integer n for which ((1+i)/(1-i))^n=1 is (a)8 (b) 16 (c) 12 (d) None of these

If I = a^2+ b^2 + c^2 , where a and b are consecutive integers and c = ab, then √I is

The least positive integer n for which ((i-1)/(i+1))^n is a real number is (A) 2 (B) 3 (C) 4 (D) 5