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

Source: ExpertsGlobal

amanvermagmat · Accepted Answer

We should know that every 4th power of odd numbers (except 5) always ends in '1'. So basically if N is an odd number ending in 1 or 3 or 7 or 9, then every fourth power of N (N^4 or N^8 or N^12 or N^16...) will end in '1'. 5 is excluded because every positive integral power of 5 ends in 5 only, and all even numbers are also excluded because integral powers of even numbers end in even, cannot end in 1.

So if a two digit number ends in 1 or 3 or 7 or 9 and if it is multiplied by itself 4 times or 8 times or 12 times or 16 times.. it will end in '1'. And in this case the probability will become 4/10; because out of 10 possibilities for last digit (0,1,2,3,4,5,6,7,8,9) - in four cases (1,3,7,9) will fourth power end in '1'.

In each of the two statements, x is a multiple of 4(12 is also a multiple of 4), so in either case - probability of it ending in '1' will be 4/10. So each statement alone is sufficient.

Hence  D answer

hellosanthosh2k2 · Answer

Hi  amanvermagmat

Question is Probability(unit digit of (two digit number)^x = 1) ?

Statement 1: is clearly sufficient
I have one question for statement 2.
I thought, x can be 0(as 0 also multiple of 4), in which case, whatever double digit number maybe, unit digit will be always one. and probability = 1

But the question clearly mentions as the number is mulitplied itself by 'x' times, so x can't be zero, it has to be some positive integer value, which is multiple of 4(as per statement2).

So statement 2 is also sufficient.

Is my understanding correct?

amanvermagmat · Answer

Hi

Yes, my reasoning is also same as yours. As it says in the question stem that number is multiplied by itself 'x' times, so saying a number multiplied by itself '0' times doesn't make sense. Hence we reject that possibility. x has to be 1 or more. And both statements mention that its a multiple of 4, so.. D answer

hellosanthosh2k2 · Answer

Thanks  amanvermagmat

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

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