If k is a non negative integer, what is the remainder when 7

Question

If k is a non negative integer, what is the remainder when    + 3  is divided by 10 ?

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

EvaJager · Answer

In fact, the question asks what is the last digit of the number  7^{12k+2}+3 .

The powers of 7 end in 7,9,3,1 cyclically, so, because  12k+2  is a multiple of 4 plus 2,  7^{12k+2}  ends in 9.
In conclusion,  7^{12k+2}+3  ends in 2.

Answer C

mandyrhtdm · Answer

Better way, it says K is a non negative integer means the answer has to be true for all non negative integers .. weather k is 0 or 987654 ....
Put k = 0 and wrap it up!

EvaJager · Answer

What if one of the answers, say E, would have been "Cannot be determined" ?

mindmind · Answer

For finding remainders with 5 & 10 : just find the last digit of the eqn

7 has a cyclicity of 4: 7,9,3,1

= 7^{12k}*7^{2} + 3

7^2 last digit : 9

12*K = Multiple of 4 : hence lats digit : 1

(7^{12})^{K}  : last digit : 1

= 9*1 + 3

=12

=12/10

R =2

GMATBaumgartner · Answer

Hi mindmind,
I am getting units digit as 7 when i try to find  (7^{12})^{K} 
not sure where I am going wrong. can you please elaborate on that step ?

mindmind · Answer

=7^{12*k} ... 7^{0} .. 7^{12} ....7^{24} ....

= 7^{4n}  .. all of the above are multiples of 4 ....

Cyclicity of 7 :4
1 >> 7
2>>,9
3>>,3,
 4>> 1

So , the last digit is 1.

GMATBaumgartner · Answer

ok thanks, I was taking value of k=0,1,2 and getting 0,12,24 ... which on dividing by cyclicity 4 - reminder 0 .Does this also show the same as above(w/o taking it as a multiple of 4) .

jlgdr · Answer

First we have that 7^14 = (10-3)^14, so we stay with (-3)^14/10, now, since exponent is even number will be positive. Then (3^2)^7 / 10 = (10 - 1)^7/10. So we have a remainder of -1 + 3 = 2, when the expression is divided by 10.

We could also have just realized that since its division by 10, the only thing we need is the units digit as the remainder of a number when divided by 10 is the units digit. Therefore 7^14 will have UD of 9 + 3 = 12 UD of 2 / 10 remainder is 2

Hope this clarifies

Cheers
J

CrackverbalGMAT · Answer

Let us say k = 1 then 7^(14) + 3 will give what remainder. Let us try an find a pattern here in the units digit as it will be the remainder when the number is divided by 10.

7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
So the pattern repeats in the cycle of 4. At 7^12 it will be 1 and 7^14 will be 9.

9 + 3 will give us 12 hence when 12 is divided by 10 the remainder is 2.

jlgdr · Answer

Please check the solutions using the units digit posted above. They are more straightforward. Remember than when a number is divided by 10 all we care is the units digit which will be the remainder

Thanks
Cheers
J

Posted from my mobile device

vishwaprakash · Answer

My Attempt:

This question basically tests concept of power cycle and Remainder property

7 has power cycle of 4, that means unit digit will follow below patter.
Unit digit of 7^1 -> 7 (4K +1)
Unit digit of 7^2 -> 9 (4K +2)
Unit digit of 7^3 -> 3 (4K +3)
Unit digit of 7^4 -> 1 (4K)

Cycle repeats after 4.
(12k+2)/4 will leave remainder 2 -> (4K + 2 ) patter. 
Hence 7 ^ (12k+2) wil have unit digit as 9

Hence 7^(12k+2)+3 will have unit digit as 2.

Hence anything with unit digit as 2 when divided by 10 will leave remainder as 2

Hence Option B is correct

bumpbot · Answer

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If k is a non negative integer, what is the remainder when 7

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If k is a non negative integer, what is the remainder when 7

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