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# Find the greatest integer:

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Joined: 01 Oct 2017
Posts: 3
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Kudos [?]: 2 [1] , given: 1

Find the greatest integer: [#permalink]  29 Oct 2017, 08:41
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Question Stats:

66% (01:26) correct 33% (00:00) wrong based on 3 sessions
Find the greatest integer

a) $$10^{10} + 2^{100}$$

b) $$100^{10} + 2^{10}$$

c) $$(100 + 2 ) ^{10}$$

Can anyone suggest me the efficient way to solve this question?
[Reveal] Spoiler: OA

Last edited by Carcass on 29 Oct 2017, 09:26, edited 1 time in total.
Edited by Carcass
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Joined: 01 Nov 2017
Posts: 351
Followers: 4

Kudos [?]: 98 [2] , given: 2

Re: Find the greatest integer: [#permalink]  01 Nov 2017, 19:12
2
KUDOS
Expert's post
sandesh10 wrote:
Find the greatest integer

a) $$10^{10} + 2^{100}$$

b) $$100^{10} + 2^{10}$$

c) $$(100 + 2 ) ^{10}$$

Can anyone suggest me the efficient way to solve this question?

Hi...

The easiest way would be..
Compare a and B..

a) $$10^{10} + 2^{100}$$

b) $$100^{10} + 2^{10}= (10^{10})^{10}+2^{10}=10^{100}+2^{10}$$

Now when you compare two 10^{100} is way GREATER than 2^{100} as compared to 10^{10} and 2^{10} ..

Otherwise a exact way would be...

a) $$10^{10} + 2^{100}=10^{10}+(2^{10})^10=10^{10}+1024^{10}=10^{10}+10^{30}~10^30$$

b) $$100^{10} + 2^{10}=10^{100}+10^3$$

SO B is clearly the bigger one

Now let's compare B and C

c) $$(100 + 2 ) ^{10}$$
When you expand it 100^{10}+100^9*2^1+.....+2^10

So here C is bigger..

Also

b) $$100^{10} + 2^{10}~100^{10}$$

c) $$(100 + 2 ) ^{10}=102^{10}$$

C wins
_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Re: Find the greatest integer:   [#permalink] 01 Nov 2017, 19:12
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