Numeric Prob

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Karthik12: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Sun Jul 20, 2008 10:36 pm

by Karthik12 » Mon Aug 04, 2008 10:10 pm

Let X = (√3 + √5)222. If X is written in decimal form, what will be the first digit after the decimal point?

1. 5
2. 6
3. 7
4. 8
5. 9

Pls explain how to get the answer

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Source: Beat The GMAT — Problem Solving | Mon Aug 04, 2008 10:10 pm

pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Tue Aug 05, 2008 1:53 am

Anyone?

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sudhir3127: Legendary Member; Posts: 829; Joined: Mon Jul 07, 2008 10:09 pmFollowed by: 3 members; Location: INDIA; Thanked: 84 times

by sudhir3127 » Tue Aug 05, 2008 4:13 am

are you sure the question reads X = (√3 + √5)222 and not

X = (√3 + √5)^222...

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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

by parallel_chase » Tue Aug 05, 2008 6:04 am

if the question is (√3 + √5)222 then 8 will be after the decimal point.

√3 ~ 1.7
√5 ~ 2.2

(1.7+2.2)222 = 3.9*222 = 765.8

now if the question is (√3 + √5)^222 then you would probably have to use modular arithmetic.

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pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Tue Aug 05, 2008 6:49 am

Chase I did approximations and I was wrong because the digit is 9.
I did with 1 like you.
(1,7 + 2,2)*222=865.8
I did with 2 after looking at the values of these sqrt
(1,73+2,23)*222=879.12

I finally checked with excel and with the real values of sqrt 3 and 5, the result is:
880.922

I wonder if it is possible to answer this question.

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sudhir3127: Legendary Member; Posts: 829; Joined: Mon Jul 07, 2008 10:09 pmFollowed by: 3 members; Location: INDIA; Thanked: 84 times

by sudhir3127 » Tue Aug 05, 2008 7:00 am

i know for sure the answer is 6.

heres what i did..

(√3 + √5)^222
= ((sq rt 3 + sq rt 5 )^2)^111

= ( 8 +2sqrt 15)^111

we know for sure that (2sqrt 15)^111 will only give us the decimal.
(2sqrt 15)^111 will be (2sqrt 15) ( (2sqrt 15)^111

root 15 if we take only 2 decimal point is 3.8

hence 2*3.8 = 7.6

hence the answer is 6.

but then if we take more decimals our answer changes.. so i am ot very sure abt the technique...

PS: i used scientific calculator to find the answer and its indeed 6 as the answer.

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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

by parallel_chase » Tue Aug 05, 2008 7:14 am

You are absolutely right if we take more decimals the answer will change to 7.
Your method is correct, but which question is correct your question or KARTIK's question, secondly the question does not mention limit for the decimal range.

What is the OA?

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pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Tue Aug 05, 2008 7:18 am

Ok Sudhir 6 or 9 it depends on which question we work on.
Karthik can you check your question and tell the OA?

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parallel_chase: Legendary Member; Posts: 1153; Joined: Wed Jun 20, 2007 6:21 amFollowed by: 2 members; Thanked: 146 times

by parallel_chase » Tue Aug 05, 2008 7:27 am

Here is the real question and of course the answer is 6.

(√3 + √5)^222 = ((√3 + √5)^2)^111 = (3+5+2√3.√5)^111
or (8+2√3.√5)^111.
(2√3.√5)^111.
Now 111th power of √3 and √5 will leave √3 and √5 at the end .
So then just multiply 2.√3.√5 and the first decimal is 6.

This question is not from GMAT, Therefore, even if you cant get it right nothing to worry.

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sudhir3127: Legendary Member; Posts: 829; Joined: Mon Jul 07, 2008 10:09 pmFollowed by: 3 members; Location: INDIA; Thanked: 84 times

by sudhir3127 » Tue Aug 05, 2008 7:35 am

good stuff chase...

as i said .. when we multiply 2.√3.√5 and the first decimal is 6

we need to be careful abt the decimal places. as the above multiplication doesnt actually give u 6 ( its gives 7) if we take more one decimal place..

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pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Tue Aug 05, 2008 8:52 am

I don't follow how you know that
if 2.√3.√5 has a first decimal of 6
then (2.√3.√5)*(2.√3.√5)^110 has a first decimal of 6
then (8+2√3.√5)^111 has a first decimal of 6 ?
What I think is that these numbers also multiply the decimals...

Just to be sure, "the first digit after the decimal point", it talks about the tenths right?

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Karthik12: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Sun Jul 20, 2008 10:36 pm

by Karthik12 » Tue Aug 05, 2008 9:04 pm

Sorry guys....I forget to put power symbol in the question...so
X=(√3 + √5)^222

the correct ans is 9 (option 5)...

I dont know how is that.......

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Rosen_Kralev: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Thu Aug 07, 2008 11:02 am; Location: Cambridge, MA

by Rosen_Kralev » Thu Aug 07, 2008 11:13 am

Hi everyone,

(√5 + √3)^222 + (√5 - √3)^222

is an integer.

√5 - √3 is pretty small (less than 0.9, for all we care) and so you can bet that when raised to the 222nd power, it will be tiny (to be read: less than 0.1)

Have fun! I just found out about this message board by chance and don't have to go through all problems, but if there's something anyone is stuck on, please email me and I'd be more than happy to help.

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