What is the tens digit of 7^1415?

Question

A. 0
B. 1
C. 2
D. 3
E. 4

Skywalker18 · Accepted Answer

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 . (We can multiply the last 2 digits only as we care about ten's digit )
0 , 4 , 4 , 0 . 
1415= 4*353 + 3
The ten's digit will be 4 .
Answer E

chetan2u · Answer

Hi,
  Another approach to this Q, which would be correct for all numbers..

the Q can be answered if we can find the remainder when  7^{1415}  is divided by 100..
now 7^4=2401..
 
 (7^4)^{353} * 7^3 = (2401)^{353}*7^3 ..
2401 will always leave a remainder of 1 
so finally remainder =  1* 7^3 = 1* 343 ..
or remainder = 43..
 tens digit =4 
E

Manonamission · Answer

I am bit confused here

Clearly the cyclicity is 4 here. 7^1415 = 7 ^ = (7 ^4.353) . (7 ^3) = (7^4.(353)) . (7^3) = (2401^353) . ( 7^3 ) = (Tens digit of a number that ends in 1 is equal to the last digit of power multiplied by the tens digit of the number) .(7^3) = (0) . (343) = 0 = The tens digit is 0.

KarishmaB · Answer

Yes, the cyclicity is 4 but considering that we start from the exponent of 1, the cyclicity is 
0, 4, 4, 0
0, 4, 4, 0
...
etc

So 1415 = 4*353 + 3 means 353 full cycles and 3 more so you go to 0, 4,  4 
The digit will be 4.

vitaliyGMAT · Answer

7^{1415}  =  7^{1410}*7  =  49^{705}*7  =  49^{704}*49  =  01^{352}*49  =  1*49=49 
Tens digit is 4

bazu · Answer

Hi chetan2u, sorry, but why you are calculating (7^4)^{354} * 7^3? Shouldn't it be (7^4)^{353} * 7^3? Or am I missing something??

adkikani · Answer

Hi  VeritasKarishma   gmatbusters   Gladiator59   Bunuel   EMPOWERgmatRichC

I adore  chetan2u  's approach above. Is there a way to skip two digit multiplications since
they are too prone to error?

daludada · Answer

Why to complicate with the various way to approach the same.

Lets understand the rule of cyclicity!

7 has a cyclicity of 4...i.e every 4th number the same pattern repeats

Now divide the exponent 1415 by 4, the balance would be 3. 
now its just 7 the cube...i.e 7x7x7.....the unit digit can be easily found.

cyclicity rule is as below:

1 1
2 4
3 4
4 2
5 1
6 1
7 4
8 4
9 2
10 1

Apply the above yardstick and solve it accordingly.

Fdambro294 · Answer

When we divide a number by 100, the last 2 digits of the Dividend will be the Remainder.

Well, let’s try it out with the Chinese Remainder Theorem..... (not applicable on GMAT)

(7)^1415 /100 ———> Rem of = ?

Split up the denominator into: 50 * 2 = 100

(7)^1415 /50

(7^2)^707 * (7) /50

(49)^707 * (7) /50

(50 - 1)^707 * (7) /50 ——> Rem of = ?

Using the Concept of the Binomial Theorem, when we multiply out (50 - 1)^707 ——-> every term will be divisible by 50 except the last term of: (-1)^707 = -1

Excess remainder therefore =

(-1) * 7 = -7

-7 + (divisor of 50) = 43

n = 50a + 43

The given number has the characteristic that when it is divided by 50 ——-> yields a remainder of +43

AND

(7)^1415 /2 ——-> since the result will be Odd, it will yield a remainder of 1 when divided by 2

N = 2b + 1

And

N = 50a + 43

So, we are looking for the first ODD number that, when divided by 50, will yield a remainder of 43

That number is 43

Thus:

(7)^1415 when divided by 100 will yield a remainder of ——-> 43

The last 2 digits must be 43

And the tens digit must be 4

E

Posted from my mobile device

CEdward · Answer

Can we use this same approach for :
a) any exponent
b) questions that ask for digits larger than tens ( e.g. hundreds )

Posted from my mobile device

BrushMyQuant · Answer

We need to find the tens digit of  7^{1415}

Last Two digits of 7 follow following pattern

Last Two digits of 7^1 = 07 
 Last Two digits of 7^2 = 49 
 Last Two digits of 7^3 = 49*7 = 43 
 Last Two digits of 7^4 = 43*7 = 01 
 Last Two digits of 7^5 = 01*7 = 07 
 Last Two digits of 7^6 = 07*7 = 49

=> We have a cycle of 4 
 => 1415 = 1412 + 3 
 =>  7^{1415}  will have the same last two digits as  7^{3}  = 43 
 => Tens' digit = 4

So, Answer will be E 
Hope it helps!

Link to Theory for Last Two digits of exponents here    .

Link to Theory for Units' digit of exponents here    .

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What is the tens digit of 7^1415?

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What is the tens digit of 7^1415?

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