A two digit number is randomly selected and multiplied by itself x

Question

A two digit number is randomly selected and multiplied by itself x times. What is the probability that the unit’s digit of the resulting number is 1?
 
(1) x = 12.
(2) x is a multiple of 4.

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sujoykrdatta · Answer

Any number raised to the 4th power or a power that is a multiple of 4 has the same units digit as n

Keeping that in mind, both statements provide the same information that x is a multiple of 4 (note that x cannot be 0 in statement 2)

To get the units digit as 1 when any number is raised to the 4th power (or a power that is a multiple of 4), the units digit of the original number must be 1, 3, 7 or 9

Since we can definitely find how many 2-digit numbers are there that end in 1,3,7 or 9, we can determine the probability as well (there are 36 such numbers out of 90 such 2-digit numbers => prob = 36/90 = 0.4)

Each statement separately sufficient (D)

Dereno · Answer

Let the two digit number be n.

When the number is multiplied x times = n*n^x = n ^(x+1)

The total number of two digit numbers = 99-10+1 = 90.

We need to find : What is the probability that the unit’s digit of the resulting number is 1?

Cyclicity of numbers:

Unit digit as 1: a1*a1 multiplied any times will yield unit digit as 1.

Unit digit as 3: cyclicity of 4 = 3*3*3*3= _ _ 1.

Unit digit as 7: Cyclicity of 7 = 7*7*7*7 = _ _ 1.

Unit digit as 9: Cyclicity of 2 = 9*9 = _ _ 1.

Statement 1:

x = 12.

Then, n^(x+1) = n^13

The only cyclicity that matches this is when unit digit is 1.

10 such numbers exists = (11,21,31,41,51,62,71,81,91)

Probabilty = 9/90 = 1/10.

Hence,  Sufficient

Statement 2:

x is a multiple of 4.

Then, n ^(4k+1) .

The only cyclicity that matches this is when unit digit is 1.

10 such numbers exists = (11,21,31,41,51,62,71,81,91)

Probabilty = 9/90 = 1/10.

Hence,  Sufficient

Option D

ExpertsGlobal5 · Answer

Explanation Video:

https://www.youtube.com/watch?v=ly3IZC1wuCE

sequimaxime · Answer

A really good question on units digit , Lets solve it with basics of what we learn abiut unit digit
let ab be the 2 digit number so (ab)^x unit digit is what we trying to find. We need to focus on b only no need for a .
If we see cyclic format of numbers b=0,1,2,3,4,5,6,7,8,9
raising there powers we get 
0-0-0-0
1-1-1-1
2-4-6-8
3-9-7-1
4-6-4-6
5-5-5-5
6-6-6-6
7-9-3-1
8-4-2-6
9-1-9-1
and they repeat

Lets consider statement 1
x=12
multiple of 4 hence , we get the probability 4/10 = 2/5

lets consider statement 2
x = multiple of 4 - still we get the same results

Hence  D Each alone is sufficient.

A two digit number is randomly selected and multiplied by itself x

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GMAT Data Sufficiency (DS) Questions

A two digit number is randomly selected and multiplied by itself x

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