Polynomials related Questions by Balaji

[Balaji]

A 4 digit prime no when divided by 24 leaves a remainder r. how many possible values can r take ?



a)7



b)8



c)9



d)10

answer choice b



solution



24k+r=prime

r is not equal to 0

r can be 1

if r=2 then 24k+r will multiple of 2

24k+3=3(8k+1)

r has to be coprime with 24

r can be 1,5,7,11,13,17,19,23



8 values totally nothing pi (24)


 

 

[Balaji]

Give the 4 two digit factors of 7^100 - 5^100 ?


a. 37,74,16,48



b. 25,50,33,40



c. none of the above 



d. cannot be determined

answer choice a



solution



dividend= (7^2)^50 - (5^2)^50



dividend=(7^2)^25-(5^2)^25



dividend=(2401)^25-(625)^25 



this will divisibly by 2401-625=1776



1776= 2*2*2*2*3*37



hence 37,74,16,48


 

[Balaji]

what is the highest power of 24 that can divide 70! ?



a. 20



b. 22



c. 25



d. 33

answer choice b



solution



highest power of 24 that divides 70!

24=2^3*3



highest power 3 in 70!



70/3

=23+ quotients

=23+7+2+1

=33



(2^2*3)^22=2^66*3^22 divides 70!



24^22 divides 70!