What is the tens digit of the number 2007^2007?

Question

What is the tens digit of the number  2007^{2007} ?

A. 0
B. 2
C. 4
D. 6
E. 8

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sumit99kr · Answer

For tens digit one needs to find the remainder when divided by 100.

Since 2007 has last digit as 7, we will focus on 7.
7 has a cyclicity of 4.

2007 = 4n+3, 
Now 2007^(4n+3), We need to find the remainder for 7^(4n+3)/100.
Using remainder theorem, 
7^4n/100= rem 1 & 7^3/100=343/100 rem 43.

Thus the overall remainder is 43x1=43.

Tens digit is 4.

Hence C is the correct choice.

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gmatophobia · Answer

We can write few terms to see if there exists a pattern

2007^1  ends with 07

2007^2  ends with 49  (We need to multiply the last two digits 07 * 07)

2007^3  ends with 43  (We need to multiply the last two digits 49 * 07)

2007^4  ends with 01  (We need to multiply the last two digits 43 * 07)

2007^5  ends with 07  (We need to multiply the last two digits 01 * 07)

So, we see the pattern repeating after 4 terms

2007^{2007} = 2007^{2004} * 2007^{3}

As the cyclicity is 4, the last two terms of  2007^{2007}  will be 43

Tens Digit = 4

Option C

mithunkrishna · Answer

which remainder theorem did you use to arrive at remainder of (7^4m)/100=1?

A_Nishith · Answer

We can write few terms to see if there exists a pattern

2007^1 ends with 07 ( Trick  : since last two digit is decided by last two numbers as Zeros is in between)

2007^2 ends with 49 (We need to multiply the last two digits 07 * 07)

2007^3 ends with 43 (We need to multiply the last two digits 49 * 07)

2007^4 ends with 01 (We need to multiply the last two digits 43 * 07)

2007^5 ends with 07 (We need to multiply the last two digits 01 * 07)

So, we see the pattern repeating after 4 terms

So,  Power of 2007 can be written as (4*501 + 3) . 
=> 2007^3 decides the last Ten's Digit. 
We have seen  2007^3 ends with 43 .

As the cyclicity is 4, the last two terms of 2007^2007
will be 43

Tens Digit = 4

BrushMyQuant · Answer

We need to find the tens digit of  2007^{2007}

Last Two digits of 2007 = 07 
Last Two Digits of 07 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 
 => 2007= 2004 + 3 
 =>  2007^{2007}  will have the same last two digits as  7^{2007}  = will have the same last two digits as  7^{3}  = 43 
 => Tens' digit = 4

So, Answer will be C 
Hope it helps!

Link to Theory for Last Two digits of exponents here    .

Link to Theory for Units' digit of exponents here    .

What is the tens digit of the number 2007^2007?

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What is the tens digit of the number 2007^2007?

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