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Math Expert V
Joined: 02 Sep 2009
Posts: 59020
What is the tens digit of 7^241?  [#permalink]

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4 00:00

Difficulty:   85% (hard)

Question Stats: 58% (01:25) correct 42% (01:33) wrong based on 257 sessions

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What is the tens digit of $$7^{241}$$?

A. 0
B. 2
C. 4
D. 6
E. 7 This question was provided by Math Revolution for the Game of Timers Competition _________________
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What is the tens digit of 7^241?  [#permalink]

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IMO : A

What is the tens digit of

7^241 ?

A. 0
B. 2
C. 4
D. 6
E. 7

Cyclicity of 7 follows 4 cycles,

7,49,343,2401, but it is interesting that the cyclicity of the tens digit also follows the same pattern. so the tens digit will be like

..07,..49,.....43,....01
This is because we multiply the tens digit by 7 as well and the powers are carried over to the 100s digit and so on.

So the tens digit of 7^241 will be 0 from (...07)

A:0
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What is the tens digit of 7^241?  [#permalink]

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Tens digit cycle
7 ^ 1= 07
7 ^ 2=49
7 ^ 3=43
7 ^ 4=01
7 ^ 5=07
.
.
.
and so on it repeats in interval of 4 241 is 4k+1
therefore tens digit is 0.

Originally posted by manass on 23 Jul 2019, 08:22.
Last edited by manass on 24 Jul 2019, 08:06, edited 1 time in total.
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What is the tens digit of 7^241?  [#permalink]

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Quote:
What is the tens digit of 7ˆ241?

A. 0
B. 2
C. 4
D. 6
E. 7

Cycles of 7 are:
7ˆ1=07
ˆ2=49
ˆ3=43
ˆ4=01
ˆ5=07… after 4 it repeats pattern 0,4,4,0;

So the remainder of 241/4 is 1, which gives a tens of 0.

Originally posted by exc4libur on 23 Jul 2019, 08:25.
Last edited by exc4libur on 23 Jul 2019, 09:35, edited 1 time in total.
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What is the tens digit of 7^241?  [#permalink]

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0 has to be the answer.
the cyclicity of 7 is 4 hence 1 will be the remainder with 241.

hence 0 will be the tens digit.
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Originally posted by Prasannathawait on 23 Jul 2019, 08:26.
Last edited by Prasannathawait on 23 Jul 2019, 10:44, edited 1 time in total.
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Re: What is the tens digit of 7^241?  [#permalink]

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1

cyclicity of 7 with respect to tens digit is 0,4,4,0,0,4,4...and so on
every 4th and 5th power is 0.
imples, 7 raise to 240 tens digit is 0 and 7 raise to 241 tens digit is 0
So A
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Re: What is the tens digit of 7^241?  [#permalink]

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What is the tens digit of $$7^{241}$$
As of 7 follows a cyclic power of 4
thus uni digit
$$7^1$$= 7
$$7^2$$ = 49
$$7^3$$ = 343
$$7^4$$ = 2301

now $$7^5$$ will be 16107
241 leaves a remainder 1 thus unit digit will same as $$7^1$$ and we know previous number would have given 01 and thus no carry forward
Hence tens digit will be 0
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Re: What is the tens digit of 7^241?  [#permalink]

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This is the cyclical property of numbers.

For number 7, the property is as such:
1st - 7
2nd - 9
3rd - 3
4th - 1
5th - 7 (it repeats from above)

Hence there are 4 cycles.
241 divided by 4 has a remainder of 1. Hence you take the first cycle above, which is 7.
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What is the tens digit of 7^241?  [#permalink]

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We can extract the tens digit by finding the last two digits of $$7^{241}$$
The last two digits of $$7^{241}$$ = Remainder when $$7^{241}$$ is divided by 100.

Thus, we need to find the Remainder of $$\frac{7^{241}}{100}$$

Finding the remainder using Euler’s Theorem:
We will convert $$7^{241}$$ into a smaller and “similar” number which will give the same remainder as $$7^{241}$$ when divided by 100.

Converting $$7^{241}$$ into a smaller and similar number:
For that, we will first find Euler’s Number of the denominator-
Mathematically, the Euler number of a number z denoted by the symbol E(z) is calculated as explained below.
$$E(z) = z * [1 – \frac{1}{P}]*[1 – \frac{1}{Q}]$$
where P, and Q are the different prime factors of z.

Therefore, $$E(100) = 100 * [1 – \frac{1}{2}] * [1 – \frac{1}{5}] = 100 * \frac{1}{2} * \frac{4}{5} = 40$$-> (a)

The smaller and similar number will have a base equal to Remainder($$\frac{7}{100}$$) = 7 -> (b)
And it will be raised to the power of Remainder($$\frac{241}{Euler’s number of 100}$$) = Remainder of $$\frac{241}{40}$$ = 1 -> (c)

Therefore from (b) and (c), the smaller and “similar” number which will give the same remainder as $$7^{241}$$ when divided by 100 is $$7^1$$ = 7 -> (d)

Therefore the last two digits of $$7^1$$ = Remainder of $$\frac{7^1}{100}$$ = Last two digits of $$7^{241}$$ = 07 -> (e)

Hence from (e), the tens digit of $$7^{241}$$ is 0

Originally posted by Sayon on 23 Jul 2019, 08:37.
Last edited by Sayon on 23 Jul 2019, 22:26, edited 1 time in total.
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What is the tens digit of 7^241?  [#permalink]

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Quote:
What is the tens digit of 7^241?

A. 0
B. 2
C. 4
D. 6
E. 7

7 to any power has tens & units digits as 07,49,43,01

Now 241/4, remainder is 1
And 7 to power 1 --> tens digit is 0
Hence A
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Originally posted by kitipriyanka on 23 Jul 2019, 08:38.
Last edited by kitipriyanka on 23 Jul 2019, 08:45, edited 1 time in total.
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Re: What is the tens digit of 7^241?  [#permalink]

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What is the tens digit of 72417241?

A. 0
B. 2
C. 4
D. 6
E. 7

cyclicity of 7
7^1 ; 7
7^2 ; 9
7^3; 3
7^4; 1
for 7^241 ; we get ; 7 as units digit and 0 as tens digit
IMO A
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What is the tens digit of 7^241?  [#permalink]

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7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649

We should see this as pattern recognition . We have a cycle of 4:

0 , 4 , 4 , 0

241= 4*60 + 1

The ten's digit will be 0

Originally posted by Mizar18 on 23 Jul 2019, 08:42.
Last edited by Mizar18 on 23 Jul 2019, 18:24, edited 1 time in total.
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What is the tens digit of 7^241?  [#permalink]

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Cycle of 7 is 07,49,43 and 01. 241 gives remainder 1 when divided by 4 hence 07 is the value. The ans is A

Originally posted by Shreshtha55 on 23 Jul 2019, 08:43.
Last edited by Shreshtha55 on 23 Jul 2019, 08:44, edited 1 time in total.
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Re: What is the tens digit of 7^241?  [#permalink]

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7 has a unit digit cyclicity of 4: 7,9,3,1. So, divide 241 by 4 and we get the remainder 1. Hence, the answer is 7.

P.s. if the no. was divisible by 4, then the answer would have been 1 (the 4th digit in the cycle).
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Re: What is the tens digit of 7^241?  [#permalink]

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What is the tens digit of 7^241?

This question tests your ability for pattern recognition
so let's analyze the question
7^1 = 7
7^2 = 49
7^3 = 343
7^4= 2401
7^5= 16807
7^6=117649
7^7=823543

There you go we see a pattern 44, 00, 44, 00 ...... and it continues. Now all you have to do is to check where 241 falls. Every 4 & 5th number is a zero. 241 = 240+1 Which means 5. Hence the answer is 0.

A. 0
B. 2
C. 4
D. 6
E. 7

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What is the tens digit of 7^241?  [#permalink]

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What is the tens digit of $$7^2^4^1$$?

A. 0
B. 2
C. 4
D. 6
E. 7

There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition.

The tens digit of 7 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 4: {4, 4, 0, 0 }:
The tens digit of 7^2=49 is 4;
The tens digit of 7^3=343 is 4;
The tens digit of 7^4=...01 is 0 (how to calculate: multiply 43 by 7 to get ...01 as the last two digits);
The tens digit of 7^5=...07 is 0 (how to calculate: multiply 01 by 7 to get ...07 as the last two digit);
The tens digit of 7^6=...49 is 4 (how to calculate: multiply 07 by 7 to get ...49 as the last two digits);
The tens digit of 7^7=...43 is 4 (how to calculate: multiply 49 by 7 to get ...43 as the last two digits).
The tens digit of 7^8=...01 is 4 (how to calculate: multiply 43 by 7 to get ...01 as the last two digits).
The tens digit of 7^9=...07 is 4 (how to calculate: multiply 01 by 7 to get ...07 as the last two digits).

In general, the tens digit of $$7^(^4^x^+^1^) = 0$$
241 = 4*60 + 1
Tenth digit of $$7^2^4^1$$ = 0

IMO A
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Originally posted by Kinshook on 23 Jul 2019, 08:49.
Last edited by Kinshook on 23 Jul 2019, 08:51, edited 1 time in total.
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Re: What is the tens digit of 7^241?  [#permalink]

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7^4 = 2401

7^ 241 = (7^4)^60 . 7
(2401)^60.7

A Bit theory
Find the 10 digit of 25x25 = 625 ==> 2

so if we divide 625 by 100 then take remainder
Remainder is 25
the tens digit is 2

On the same line divide (2401)^60.7 by 100

Remainder = 1X 7 = 7 --> 07

Tens digit is 0

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Re: What is the tens digit of 7^241?  [#permalink]

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What is the tens digit of 7^241?

If we just multiply 7*7*7... we will see the pattern.
1. 07
2. 49
3.343
4. 2401

it has cyclicity 4 and all we need just divide 241/4= 60 and remainder is 1.
in power 1, 7 has ten digit zero

IMO A
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Re: What is the tens digit of 7^241?  [#permalink]

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7^1 = 07
7^2 = 49
7^3 = 43
7^4 = 01
7^5 = 07
7^6 = 49 so on .....

7^241 = 7^(4*60 + 1) = 07

IMO Option A

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Re: What is the tens digit of 7^241?  [#permalink]

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Tens digit follows the cycle of 0440 : so divide 241/4 leaves a remainder of 1,
In cycle , it makes 0 as on 5th place.
So IMO A : Re: What is the tens digit of 7^241?   [#permalink] 23 Jul 2019, 09:05

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