The largest number amongst the following that will perfectly

Question

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) 100,000,00

GMAT TIGER · Answer

probably can also be solved by factorization.

but my logic is:
101^1 - 1 = 100
101^2 - 1 = 10201 - 1 = 10200
101^3 - 1 = 1030301 - 1 = 1030300

all are divided by 100.
so A.

any body has more succint way?

walker · Answer

I'm between A and C. Don't know how to exclude C....

incognito1 · Answer

I'm going to take a stab at this and go with (B)

101^1 = 101
101^2 = 10201
101^3 = 1030301
.
.
101^10 = would end with 1001

Similarly, 101^100 would end with 10001

It follows that 101^100 - 1 (equivalent to N.....10,001 - 1) would be divisible by (b)

kevincan · Answer

(100 + 1)^100 - 1 = 100^100 + ... + 100((100)^1)((1)^99) + 1^100 - 1

This is a multiple of 10,000

walker · Answer

My reasons are the same, but

(100 + 1)^{100} - 1 = 1^{100}+99*1^{99}*100^1+\frac{99*98}{2}*1^{98}*100^2+....+99*1^1*100^{99}+100^{100} -1=

=99*100+\frac{99*98}{2}*1*100^2+....+99*1*100^{99}+100^{100}=

=99*100+100^2*(\frac{99*98}{2}+....+99*100^{97}+100^{98})

So, (100 + 1)^{100} - 1 is not divisible by 10000

FN · Answer

OA is B..

here is how..

101^2 -1 =10201-1 =10200 this is divisible by 100..correct..

101^10 -1=102000-1= which is divisible 1000

101^100 - 1 will be divisible by 10,000.

GMATBLACKBELT · Answer

Why not A as well?

If it must divide by B then surely it will divide by A

FN · Answer

its asking for the largest number... comeon no +1 for me???

walker · Answer

101^100-1 =
= 11574001359990451186283141070066493936597109898321600750632049572420389134812037149366326601286302964
9578890272024960082993285013905624357430362636553574489997143361936192994493701723942231042303500000 :shock:

but I've not found my mistake...... :roll:

walker · Answer

Ahh... silly mistake...Kevincan, you are right! Should be 100 instead of 99...... :evil:

FN · Answer

this is the OE...excellent work ico

walker · Answer

by the way, if I correctly calculated, D should be the answer....

walker · Answer

Sorry, guys. It is 
27048138294215260932671947108075308336779383827810027768902010491171015143067392794394560143467445909
7335651375483564268312519281766832427980496322329650055217977882315938008175933291885667484249510000