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what are the last 2 digits of 7^2800?

 Nov 20, 2014

Best Answer 

 #3
avatar+21869 
+8

what are the last 2 digits of 7^2800  or what is $$7^{2800} \mod 100$$ ?

$$7^{2800} \mod 100 \quad | \quad 2800 = 2^4*175\\
= 7^{ 16 * 175 } \mod 100 \\
= ( 7^{16} )^{175} \mod 100 \quad | \quad 7^{16}= (7^4)^4\\
= ( (7^4)^4 )^{175} \mod 100 \quad | \quad 7^{4}= (7^2)^2\\
= ( ( (7^2)^2 )^4 )^{175} \mod 100 \quad | \quad 7^2 = 49\\
= ( ( (49)^2 )^4 )^{175} \mod 100 \quad | \quad 49^2 = 2401\\
= ( ( 2401 )^4 )^{175} \mod 100 \quad | \quad 2401 \mod 100 = 1 \quad\textcolor[rgb]{1,0,0}{ x^{a*b} \mod d = (x^a\mod d)^b \mod d }\\
= ( ( 1 )^4 )^{175} \mod 100 \quad | \quad 1^4 = 1 \\
= ( 1 )^{175} \mod 100 \quad | \quad 1^{175} = 1 \\
= 1 \mod 100 \\\\
7^{2800} \mod 100 = 1 \mod 100 \quad | \quad 1 \mod 100 = 01$$

the last 2 digits of 7^2800 = 01

 Nov 21, 2014
 #1
avatar+17746 
+8

The last digit of 7^x rotates among 1, 7, 9, and 3 because:

7^0  =    1     7^4 = 7^4 x 7^0 = 2401           7^8 = 7^4 x 7^4 x 7^0 = 5764801      ...

7^1  =    7     7^5 = 7^4 x 7^1 = 16707         7^9 = 7^4 x 7^4 x 7^1 = 40353607    ...

7^2  =  49     7^6 = 7^4 x 7^2 = 117649        ...

7^3 = 343     7^7 = 7^4 x 7^3 = 823543        ...

What I'm trying to show is extra factors of 7^4 doesn't change the last digit of the answer.

So divide the exponent by 4,

if the remainder is 0, the last digit will be a 1

if the remainder is 1, the last digit will be a 7

if the remainder is 2, the last digit will be a 9

if the remainder is 3, the last digit will be a 3

Dividing the exponent of 2800 by 4, gives a remainder of 0, so the last digit will be a 1.

 Nov 20, 2014
 #2
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Thank you for the response. What you answered is something i already know... my question is to find the last 2 digits and not just the last digit. So essentially, when 7^2800 is divided by 100, it will give the required answer.

 Nov 20, 2014
 #3
avatar+21869 
+8
Best Answer

what are the last 2 digits of 7^2800  or what is $$7^{2800} \mod 100$$ ?

$$7^{2800} \mod 100 \quad | \quad 2800 = 2^4*175\\
= 7^{ 16 * 175 } \mod 100 \\
= ( 7^{16} )^{175} \mod 100 \quad | \quad 7^{16}= (7^4)^4\\
= ( (7^4)^4 )^{175} \mod 100 \quad | \quad 7^{4}= (7^2)^2\\
= ( ( (7^2)^2 )^4 )^{175} \mod 100 \quad | \quad 7^2 = 49\\
= ( ( (49)^2 )^4 )^{175} \mod 100 \quad | \quad 49^2 = 2401\\
= ( ( 2401 )^4 )^{175} \mod 100 \quad | \quad 2401 \mod 100 = 1 \quad\textcolor[rgb]{1,0,0}{ x^{a*b} \mod d = (x^a\mod d)^b \mod d }\\
= ( ( 1 )^4 )^{175} \mod 100 \quad | \quad 1^4 = 1 \\
= ( 1 )^{175} \mod 100 \quad | \quad 1^{175} = 1 \\
= 1 \mod 100 \\\\
7^{2800} \mod 100 = 1 \mod 100 \quad | \quad 1 \mod 100 = 01$$

the last 2 digits of 7^2800 = 01

heureka Nov 21, 2014

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