f(x) denotes the remainder when x is divided by 10. For example, f(5)

Question

f(x)  denotes the remainder when  x  is divided by  10 . For example,  f(5) = 5, f(92) = 2  and  f(271) = 1 . What is the value  f(2^{2006}) + f(3^{2006}) + f(5^{2006}) + f(7^{2006}) + f(9^{2006}) ?

A.  24 
 
B.  26 
 
C.  28 
 
D.  30 
 
E.  32

AryaSwagat · Answer

Remainder when divided by 10 is the unit's digit of that number. So we have to find out unit's digit of each expression.
Let's look at the exponents of 2 starting from  2^1, 2^2, 2^3  ....... =  2,4,8 ,1 6 , 3 2 ,6 4 ,12 8 ,25 6 
clearly the unit digit have a cyclicity of 4. So to find out the unit digit of  2^{2006}  , 2006/4 leaves remainder 2, so unit digit is  4 
Similarly for  3^{2006} , 3 has also cyclicity 4, remainder of (2006/4) is 2, so unit digit is  9 .
exponents of 5 always have unit digit  5 .
7 has cyclicity 4, 2006/4 has remainder 2 so unit digit  9 .
9 has Cyclicity 2, so unit digit of  9^{2006}  is  1 .

adding all the remainders(unit digits) 4+9+5+9+1= 28.

IMO  C   
(Corrected)

Anisha1296 · Answer

Shouldn't the power be divided by the cyclicity rather than being divided by the base number?

Paras96 · Answer

f(2^2006)+f(3^2006)+f(5^2006)+f(7^2006)+f(9^2006)

We have to find the unit digit of the following,
2^2006,3^2006,5^2006,7^2006,9^2006

For, 2^2006 = (2^2004)(2^2) Taking unit digits 6*4=24, unit digit = 4
3^2006 = (3^2004)(3^2) Taking unit digits 1*9=9, unit digit = 9
5^2006 , unit digit = 5
7^2006 = (7^2004)(7^2) Taking unit digits 1*9=9, unit digit = 9
9^2006 = (9^2004)(9^2) Taking unit digits 1*1=1, unit digit = 1

Adding the unit digits from above = 4+9+5+9+1 = 28
 Hence D

gmatophobia · Answer

Note that the unit digit of each option is different. Hence, if calculate the unit digit of the sum, we can narrow down the answer

2006 = 2004 + 2

f(2^{2006})  = Unit digit = 4
 
  f(3^{2006})  = Unit digit = 9
 
  f(5^{2006})  = Unit digit = 5
 
  f(7^{2006})  = Unit digit = 9
 
  f(9^{2006})  = Unit digit = 1

Sum = 4 + 9 + 5 + 9 + 1 = Unit digit = 8

Option C

chetan2u · Answer

Remainder when divided by 10 means nothing but the unit’s digit of the number.

So we can rephrase the question as 
What is the sum of units digit of  f(2^{2006}) , f(3^{2006}) , f(5^{2006}) , f(7^{2006}) , f(9^{2006}) ?
Now 2006=4*501+2 and the units digit repeats after every 4th successive power. ( It could repeat every time, for example 1, 5, 6 and 0, or repeat after two successive powers, for example 4 and 9.)

f(2^{2006}) + f(3^{2006}) + f(5^{2006}) + f(7^{2006}) + f(9^{2006})  would mean 
 f(2^{2}) + f(3^{2}) + f(5^{2}) + f(7^{2}) + f(9^{2}) 
4+9+ 2 5+ 4 9+ 8 1 = 4+9+5+9+1 = 28

C

f(x) denotes the remainder when x is divided by 10. For example, f(5)

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f(x) denotes the remainder when x is divided by 10. For example, f(5)

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards