Given below is a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements is sufficient to answer the question. You should use the given data and your knowledge of Mathematics to choose between the possible answers.
What will be the last digit of number Xy, where X and Y are natural numbers?
I. Y leaves a remainder of 3 when divided by 8.
II. X leaves a remainder of 7 when divided by 10.
Solution
Correct Answer: Option E
From statement I:
Y leaves a remainder of 3 when divided by 8. So, Y can be 3, 11, 19, and so on. X can be any natural number.
⇒ Xycan be 23, whose last digit is 8, or it can be 33, whose last digit is 7, or there can be many other cases. Last digit cannot be uniquely found.
∴ Statement I alone is not sufficient to answer the question.
From statement II:
X leaves a remainder of 7 when divided by 10. So, X can be 7, 17, 27, and so on. Y can be any natural number.
⇒ Xycan be 73, whose last digit is 3, or it can be 72, whose last digit is 9, or there can be many other cases. Last digit cannot be uniquely found.
∴ Statement II alone is not sufficient to answer the question.
From statements I and II together:
Y leaves a remainder of 3 when divided by 8. So, Y would be of the form 8n + 3, i.e. 4(2n) + 3, where n is a whole number.
Y leaves a remainder of 7 when divided by 10. So, X would be of the form 10m + 7, where m is a whole number.
⇒ Xy = (10m + 7)( 4(2n) + 3)
We know that last digit of any natural number repeats after its every four powers.
⇒ Last digit of Xy will be same as last digit of (10m + 7)(3)
When we expand this by binomial expansion, all terms will be a multiple of 10, except 73.
So, last digit of Xy will be last digit of 73, i.e. 3.
∴ Using both the statements together, we can answer the given question.
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