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What is the remainder when 7^8 is divided by 100?

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What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 26 Jul 2018, 01:47
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[Math Revolution GMAT math practice question]

What is the remainder when \(7^8\) is divided by \(100\)?

A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)

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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 26 Jul 2018, 01:56
Typical example of big exponents, where we need to find the pattern for the last digit.

7^1=7
7^2=49
7^3 last digit will be last digit of 9x7=63 => 3
7^4 last digit will be last digit of 3x7=21 =>1
7^5 last digit will be last digit of 1x7=7 => 7
7^6 last digit will be last digit of 7x7=49 => 9
now the pattern repeats itself
7^7 last digit will be 3
7^8 last digit will be 1

Answer A
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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 26 Jul 2018, 03:32
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the remainder when \(7^8\) is divided by \(100\)?

A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)


When an integer is divided by 100, the remainder will have the same units digit as the integer.
Thus, to determine which answer choice represents the remainder when \(7^8\) is divided by 100, we need to know the units digit of \(7^8\).
When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE.

\(7^1\) --> units digit of 7.
\(7^2\) --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.)
\(7^3\) --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.)
\(7^4\) --> units digit of 1. (Since the product of the preceding units digit and 7 = 3*7 = 21.)
From here, the units digits will repeat in the same pattern: 7, 9, 3, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 7 is raised to a power that is a multiple of 4, the units digit will be 1.
Thus, \(7^8\) has a units digit of 1.


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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 26 Jul 2018, 06:21
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the remainder when \(7^8\) is divided by \(100\)?

A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)


Let's examine 7^8 - 1

Why would I do this?
Well, I know that 7^2 + 1 = 50, which is a factor of 100.
So, perhaps it's the case that 7^8 - 1 is divisible by 100, in which case 7^8 will leave a remainder of 1 when divided by 100

7^8 - 1 is a difference of squares.
So, 7^8 - 1 = (7^4 + 1)(7^4 - 1)
= (7^4 + 1)(7^2 + 1)(7^2 - 1)
= (7^4 + 1)(7^2 + 1)(7 + 1)(7 - 1)
= (7^4 + 1)(50)(8)(6)
= (7^4 + 1)(2400)
= (7^4 + 1)(24)(100)

So, we can see that 7^8 - 1 is divisible by 100
7^8 is 1 greater than 7^8 - 1, so we must get a remainder of 1 when 7^8 is divided by 100

Answer: A

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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 26 Jul 2018, 07:44
1
I did this in following way.

49^4*2^4/2^4*100
= 98^4/2^4*100
=(-2)^4/2^4
= 1
That's the answer.

Please appraise my approach and give kudos if I am correct on my way.
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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 26 Jul 2018, 11:05
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the remainder when \(7^8\) is divided by \(100\)?

A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)


\(\frac{7^4}{100}\) = Remainder 1

Thus, \(\frac{7^8}{100}\) = Remainder 1, Hence answer must be (A)
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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 28 Jul 2018, 08:14
pkd niks18 chetan2u KarishmaB gmatbusters

Quote:
When an integer is divided by 100, the remainder will have the same units digit as the integer.


I could not recall such a rule under timing pressure. Although w/o clock I figured out that if I take random number
say 202 and divide it by 100 I get remainder as 2, which is same as unit digits of 202.
My mind intuitively went into a decimal form of integers under timed stress e.g. 202/100 = 2.02 and I could not relate cyclicity to a decimal form of a fraction. Any two cents here to relate divisibility by 100 with unit digits?
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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 28 Jul 2018, 08:38
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adkikani wrote:
pkd niks18 chetan2u KarishmaB gmatbusters

Quote:
When an integer is divided by 100, the remainder will have the same units digit as the integer.


I could not recall such a rule under timing pressure. Although w/o clock I figured out that if I take random number
say 202 and divide it by 100 I get remainder as 2, which is same as unit digits of 202.
My mind intuitively went into a decimal form of integers under timed stress e.g. 202/100 = 2.02 and I could not relate cyclicity to a decimal form of a fraction. Any two cents here to relate divisibility by 100 with unit digits?


It is also true for divisibility by 10..
Reason is if the number has to be divisible by 10 or 100, it has to have 0 as units digit. So whatever on top of 0 will be units digit..
202/100= (200+2)/100=200/100+2/100..
So 200 is div and left is 2
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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 28 Jul 2018, 09:29
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adkikani wrote:
pkd niks18 chetan2u KarishmaB gmatbusters

Quote:
When an integer is divided by 100, the remainder will have the same units digit as the integer.


I could not recall such a rule under timing pressure. Although w/o clock I figured out that if I take random number
say 202 and divide it by 100 I get remainder as 2, which is same as unit digits of 202.
My mind intuitively went into a decimal form of integers under timed stress e.g. 202/100 = 2.02 and I could not relate cyclicity to a decimal form of a fraction. Any two cents here to relate divisibility by 100 with unit digits?


Hi adkikani

The basic problem that you discussed here is that you went on to think of a decimal number for a remainder question.

In most "Remainder" type question in GMAT you need not think of a decimal number. The options for these questions are also pretty straight forward integers. So as a thumb rule when you see a remainder question you should think of D=Q+R form, where R is the remainder.

and the cyclicity and properties of 10 and its powers are pretty simple to learn. You can practice few questions to internalize it
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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 29 Jul 2018, 00:58
1
adkikani wrote:
pkd niks18 chetan2u KarishmaB gmatbusters

Quote:
When an integer is divided by 100, the remainder will have the same units digit as the integer.


I could not recall such a rule under timing pressure. Although w/o clock I figured out that if I take random number
say 202 and divide it by 100 I get remainder as 2, which is same as unit digits of 202.
My mind intuitively went into a decimal form of integers under timed stress e.g. 202/100 = 2.02 and I could not relate cyclicity to a decimal form of a fraction. Any two cents here to relate divisibility by 100 with unit digits?


Another thing - remainders and decimals are two different (but equivalent of course) ways in which you can show the output of a division.

e.g.13/5 => 2 quotient and 3 remainder
or 13/5 => 2.6 (the integer part corresponds to the quotient and 6/10 = 3/5 = remainder/divisor)

So when talking about remainders, we are looking at the result of division from the quotient/remainder perspective, not from the decimal perspective.

As for the connection between units digit and division by 2/5/10 or a multiple of 10, check out the two posts I wrote for Veritas Prep:

https://www.veritasprep.com/blog/2015/1 ... questions/
https://www.veritasprep.com/blog/2015/1 ... ns-part-2/
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Re: What is the remainder when 7^8 is divided by 100?  [#permalink]

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New post 29 Jul 2018, 18:34
=>

The remainder when \(7^8\) is divided by \(100\) is equal to the final two digits of \(7^8\).
Now, \(7^1 = 7, 7^2 = 49,\) \(7^3 = 343\), and \(7^4 = 2401\).
So, the final two digits of \(7^n\) have period \(4\):
The tens digits are \(0 -> 4 -> 4 -> 0\)
and the units digits are \(7 -> 9 -> 3 -> 1.\)
It follows that the tens and units digits of \(7^8\) are \(0\) and \(1\), respectively.
Therefore, the remainder when \(7^8\) is divided by \(100\) is \(1\).

Therefore, the answer is A.
Answer : A
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Re: What is the remainder when 7^8 is divided by 100? &nbs [#permalink] 29 Jul 2018, 18:34
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