If ab is two digit positive integer, what is remainder of 8^(ab) divid

If ab is two digit positive integer, what is remainder of \(8^{ab}\) divided by 10?
To find the remainder of \(8^{ab}\) when divided by 10, unit digit of \(8^{ab}\) is to be identified.
Cyclicity of 8 is 4 i.e. unit digit of \(8^1, 8^2, 8^3\) and \(8^4\) is 8, 4, 2 and 6 respectively.

So, if ab is divisible by 4, if not, then what is the remainder when ab is divided by 4.

(1) Remainder of ab divided by 12 is 10
ab = 12k + 10 where k is any positive integer.
So,
Unit digit of \(8^{12k + 10}\) = Unit digit of \(8^{12k} \) * Unit digit of \(8^{10}\)
= 6 * 4
= 4

Hence remainder of \(8^{ab}\) divided by 10 is 4.

SUFFICIENT.

(2) Remainder of ab divided by 16 is 14
ab = 16k + 14 where k is any positive number.
So,
Unit digit of \(8^{16k + 14}\) = Unit digit of \(8^{16k} \) * Unit digit of \(8^{14}\)
= 6 * 4
= 4

Hence remainder of \(8^{ab}\) divided by 10 is 4.

SUFFICIENT.

Answer D.

We are given that ab is a positive two-digit number. We are to determine the remainder when 8^(ab) is divided by 10. In order words, we are to determine the units digit of 8^(ab).

We know that the powers of 8 have a cyclicity of 4. Since this is a data sufficiency question, we just need to establish whether the clues provided in the statements lead to a unique cycle of the power of two, i.e. 1,2,3, or 4. If we get different powers, then the statement is insufficient. In order words the remainder of ab when divided by 4 must lead to a unique value for sufficiency.

Statement 1: Remainder of ab divided by 12 is 10
Statement 1 means that ab=12k + 10 where k is a positive integer.
Since 12k is a multiple of 4, Rof(12k+10) = Rof(10)=2.
So statement 1 is sufficient, since every integral value of k will result in a remainder of 2.
Hence 8^(12+10) will always yield the same unit digits as 8^2 = 4.

Statement 2: Remainder of ab divided by 16 is 14
Similarly, statement is basicially saying ab=16k+14.
But since 16k is a multiple of 4, the remainder of ab when divided by 4 =R of (14/4) = 2.
Hence Statement 2 is also sufficient since we know that 8^(ab)=8^(16+14) = 8^2 which has unit a unit digit = 4.

The answer is therefore D.

Question is asking for unit digit of 8^(ab).

so we have to calculate reminder of ab when divided by 4.

St. 1

ab = 12n+10
reminder (ab/4) = 2
so unit digit will be 4 always.
Sufficient

St. 2
ab = 16m+14
reminder (ab/4) = 2
so unit digit will be 4 always.
Sufficient

D is answer.

\(\frac{8^{(ab)}}{10}=\frac{2^{(3ab)}}{10}\)

\(powers(2):2,4,8,16,32…=[2,4,8,6]=[4]\)

\(remainder:\frac{2^n}{10}=[2,4,8,6]\)

(1) Remainder of ab divided by 12 is 10 sufic

\(remainder:ab/12=10…ab=[10,22,34,46…]\)

\(remainder:ab/[4]=[10/4=2,22/4=2,2,2,2…]=2\)

\(remainder:\frac{2^{3ab}}{10}=\frac{2^2}{10}=4\)

(2) Remainder of ab divided by 16 is 14 sufic

\(remainder:ab/16=14…ab=[14,30,46,62…]\)

\(remainder:ab/[4]=[14/4=2,30/4=2,2,2,2…]=2\)

\(remainder:\frac{2^{3ab}}{10}=\frac{2^2}{10}=4\)

Ans (D)

cyclicity of 8 ; 8,4,2,6
given ab is a 2 digit integer
#1
Remainder of ab divided by 12 is 10
ab can be 10,22,34 ; the unit digit will always be 4 and remainder when divided by 10 ; 4
sufficient
#2
Remainder of ab divided by 16 is 14
ab ; 14,30 ; the unit digit will always be 4 and remainder when divided by 10 ; 4
sufficient
IMO D

If ab is two digit positive integer, what is remainder of 8(ab) divided by 10?

(1) Remainder of ab divided by 12 is 10
(2) Remainder of ab divided by 16 is 14