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Algebra

Cool goIITian

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22 Oct 2009 13:28:02 IST

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1.Find the remainder when 2710 +751 is divided by 10. PLEASE TELL ME THE GENERAL METHOD

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1.Find the remainder when 27¹⁰ +7⁵¹ is divided by 10.PLEASE TELL ME THE GENERAL METHOD TO THIS SUM

2.determine constant term in expansion ok (1+ x + (1/x)² + (1/x)³)¹⁰

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adi..

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22 Oct 2009 14:09:13 IST

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is the ans..for 1st one ..is 3...

Rohit Agarwal

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22 Oct 2009 14:16:07 IST

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method???

Raghav

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22 Oct 2009 14:43:08 IST

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My answer comes out to be 2(for first question).First you tell the answer,if it is correct then i will post my solution..

eragon24

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22 Oct 2009 15:20:41 IST

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eragon24

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22 Oct 2009 15:21:04 IST

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Vivek Sarda

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22 Oct 2009 15:25:56 IST

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7 powers always repeat like (last digit) 7 9 3 1 7 9 3 1 .... find the last last digit!!! ALL WE WANT IS THE LAST DIGIT

Bipin Dubey

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22 Oct 2009 16:36:17 IST

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27¹⁰ + 7⁵¹ = 3³⁰ + 7(7²)²⁵ = 9¹⁵ + 7(49)²⁵

= (10-1)¹⁵ + 7(50-1)²⁵

expand both the terms binomially, you'll get them in the form of

= (10m-1) + 7(50n-1) where m,n are integers

= 10m + 350n - 8 = 10(m+35n-1) + 2 which when divided by 10 will leave 2 as remainder

Raghav

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22 Oct 2009 18:26:37 IST

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Hint(Using number theory)

$27^{2}=-1(mod 10)$
and $7^{2}=-1(mod 10)$

eragon24

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22 Oct 2009 18:33:45 IST

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@RAGHAV bhai mainey upar KAR diya hai dekh lo using congruences

Rohit Agarwal

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22 Oct 2009 22:31:31 IST

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gimme a general method people!!

Hari Shankar

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23 Oct 2009 11:00:33 IST

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If you can learn congruences such problems are easy meat. Otherwise you usually employ a combination of Binomial Theorem and wits.

But this one doesnt need much heavy machinery. Here, as many posts have pointed out. the remainder when 27¹⁰ is divided by 10 is the same as when 7¹⁰ is divided by 10.

Now 7⁴ is of the form 10k+1. So 7¹⁰ = (7⁴) * 7² leaves a remainder of 9

Again 7⁵¹ = (7⁴)¹² * 7³ leaves a remainder of 3.

So 27¹⁰+7⁵¹ leaves a residue of 12 i.e. a remainder 2 when divided by 10.

akl

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1 Nov 2009 16:04:24 IST

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can anyone explain the theory of congruences...as done by deepak above

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