divisibilty

[gmatmachoman]

The largest number amongst the following that will perfectly divide {(101)^100} - 1 is


A.100
B.10,000
C.100100
D.100,000
E.1000

[shashank.ism]

The largest number amongst the following that will perfectly divide {(101)^100} - 1 is


A.100
B.10,000
C.100100
D.100,000
E.1000

This question has been discussed in the post :
https://www.beatthegmat.com/largest-numb ... 15519.html

[vijay_venky]

Actually this method could be solved by using the binomial expansion as well.

(a+b)^n=nc0*a^n.b^0+nc1*a^n-1*b^1+.......+ncn-1*a^1*b^n-1+ncn*a^0*b^n.

Going by this expansion,the 101^100 expansion looks like

100c0*100^100+100c1*100^99+......+100c99*100+100c100.

101^100-1 means the last digit of 101^100 that is 100c100 has to be deleted.
Let us look at the least number,(the number with least number of zeros). Because all other numbers are of higher powers pf 100, 100c99.100 should be the least number. This is 10000. So this expansion contains 4 zeros at the end and so divisible by 10000.

Typical CAT problem.

[komal]

The largest number amongst the following that will perfectly divide {(101)^100} - 1 is


A.100
B.10,000
C.100100
D.100,000
E.1000

101^2 = 10201. 101^2 - 1 = 10200. This is divisible by 100. Similarly try for 101^3 - 1 = 1030301 - 1 = 1030300.

So you can safely conclude that (101^1 - 1) to (101^9 - 1) will be divisible by 100 & then (101^10 - 1) to (101^99 - 1) will be divisible by 1000 & therefore (101^100 - 1) will be divisible by 10,000.

(B) is the answer