If 3^10–n is divisible by 4, which of the following could be the value

Question

If  3^{10}–n  is divisible by 4, which of the following could be the value of an integer n?

I. 0 
II. 1 
III. 5

A. I only 
B. II only 
C. III only 
D. II and III only 
E. I, II and III

chetan2u · Accepted Answer

1) choices
The choices give you the answer..
3^10-n
I. If n is 0, we are left with 3^10, whi h is ODD, so not div by 4
Ii. Now 1 and 5 have a difference of 4 within themselves so either both are possible or none of the two

In choices D gives both and there is no choice giving None of the above 
So D

2) cyclicity
3 gives a remainder 3
3^2=9 gives a remainder 1
3^3 =27 gives a remainder 3
3^4 =81 gives a remainder 1..
So even power leave a remainder 1
So 3^10 will leave a remainder 1 so n can be 1
Now 5 also leaves a remainder 1 so 5 can also be the answer..
D

Nikhil · Answer

Is it  3^{10-n}  or  3^{10}-n

rever08 · Answer

chetan2u  could you tell what is wrong with my reasoning.

3 power 10 would end in 0 - 5 would give another odd number. Hence only divisive by n=1

Posted from my mobile device

chetan2u · Answer

3*10 would end in 0
But 3^10=3*3*3*3*...10times
Cyclicity
3^1 ends in 3
3^2 =9 ends in 9
3^3 = 27 ends in 7
3^4=27*3 ends in 1
3^5=81*3 ends in 3 and the cyclicity happens after every 4th number
So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9..
So 3^anything positive will always be ODD

rever08 · Answer

Argh..what was I thinking?? 
Thanks mate.

GyMrAT · Answer

Given  3^{10}–n  =  4k

I.  n = 0 , we get  3^{10}–n  =  3^{10}  =  3^{4}*3^{4}*3^{2}

When  3^{4}*3^{4}*3^{2}  is divided by  4 , we get remainder as  1 . Hence Not Divisible.

n = 0  is not possible.

II.  n = 1 , we get  3^{10}–n  =  3^{10}–1

we know from above  3^{10}  divided by  4  leaves a remainder of  1

&  1  divided by  4  will leave a remainder of  1 .

Hence  3^{10}–1  divided by 4 will leave a remainder  (1 - 1) = 0 . Hence divisible.

n = 1  is possible.

III.  n = 5 , we get  3^{10}–n  =  3^{10}–5

we know from above  3^{10}  divided by  4  leaves a remainder of  1

&  5  divided by  4  will leave a remainder of  1 .

Hence  3^{10}–5  divided by 4 will leave a remainder  (1 - 1) = 0 . Hence divisible.

n = 5  is possible.

Answer D.

Thanks,
GyM

CAMANISHPARMAR · Answer

If you don't subtract anything than 3^10-0 will be odd hence option 1 and 5 can be eliminated immediately.

Now 3^10 is a big number, say x. If either 1 or 5 is subtracted from a big number the both the numbers will be divisible by 4 if either one of them is divisible by 4. (e.g. 9-1 = 8 & 9-5=4; BOTH 8 & 4 are divisible by 4)

Since there is no option which says none of these then we can safely select option D with doing any calculations!! Hope this out of the box thinking was helpful & time saving

Madhavi1990 · Answer

3^10 - n/ 4 = Q

So 3's cycle of power is 3,9,7,1...if we raise 3 to 10 times we will get a number that ends in XX9.
So 9 - 5 = 4 = divisible
9 - 8 = divisible.

So answer is II,III which is D

abhishekdadarwal2009 · Answer

3=3
3*3 =9
3*3*3=27
3*3*3*3=81

then the last digit repeats itself.
 so starting from 3 count 10 times we come to 9(last digit)

then we know that the last digit for 3^10 is 9.
and this -n is divisible by 4.that means n will be 1 or 5.

Mansoor50 · Answer

can we reason as follows:

since 3^10 is odd, 3^10 -1 is even (odd-odd gives even). and since we get an even result, it could be divisible either by 4 or 5 ?

thanks

Kaczet · Answer

3^10=729^2= 29*29 = a number ending with 41. Therefore:
1) 41-0 = 41 not divisible by 4
2) 41-1 = 40 divisible by 4 
3) 41-5 = 36 divisible by 4

walker · Answer

3^10 = (4-1)^10 = 4k+1, where k is an integer.
So any n = 4m+1 will work, where m is an integer. Only 1 and 5 can be represented as 4m+1.

Posted from my mobile device

Abhishek009 · Answer

\frac{3^{10} – n}{4}  = 0

Or,  \frac{3^2*3^2*3^2*3^2*3^2 – n}{4}  = 0

Or,  \frac{9}{4}*\frac{9}{4}*\frac{9}{4}*\frac{9}{4}*\frac{9}{4}-\frac{n}{4}  = 0

Or,  \frac{81}{4}*\frac{729}{4}-\frac{n}{4}  = 0

Or,  \frac{59049}{4}-\frac{n}{4}  = 0

Or,  \frac{59049 - n }{4}  = 0

Now, chck the options only (II) and (III) fits in perfectly, Answer must hence be (D)

deveshj21 · Answer

chetan2u 
i could not fully comprehend the point (ii) of your explanation. Can you please illustrate using a example and can it be used with other expressions also for eg 5^11/4?

Basshead · Answer

The exponents of 3 follow this pattern in the units digit = 3, 9, 7, 1

3^10 = units digit of 9.

Lets take a look at the options:

I. 0; 9 - 0 is not divisible by 4. OUT.
II. 1; 9 - 1 is divisible by 4. In.
III. 5; 5 - 1 is divisible by 4. In.

Answer is D.

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If 3^10–n is divisible by 4, which of the following could be the value

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If 3^10–n is divisible by 4, which of the following could be the value

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