Select Page

Competency in Focus: Number System

This problem from Indian Statistical Institute (ISI Entrance 2012) is based on Number System. It includes finding the remainder when a number is divided by another digit.

Next understand the problem

The last digit of  $9!+3^{9966}$ is (A) 3 (B) 9 (C) 7 (D) 1
Source of the problem
Indian Statistical Institute (ISI) 2012 Problem 8.

Number system

4/10
Suggested Book
Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

Do you really need a hint? Try it first!
Unit digit of the whoe expression will be sum of unit digit of the first term and unit digit of the second term. So if the first term gives last digit 5 and 2nd terms gives 2 then unit digit of whole expresion is (5+2) or 7.
when we expand $9!$ there will be 5 and 2 in between that when multiplied will give 10 as a factor so the term $9!$ will have $0!$ as last digit.
$3^{1}$ has last digit as 3 $3^{2}$ has last digit as 9 $3^{3}$ has last digit as 7 $3^{4}$ has last digit as 1 $3^{5}$ has last digit as 3  and the pattern repeats with the power(indxe) at an interval of 4.
9966 when divided by 4 gives 2 as remainder. So, in $3^{9964} 3^{2}$ , 9 will be the last digit.

I.S.I. & C.M.I. Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are: B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.
The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

Balls-go-round |ISI MStat PSB 2013 Problem 10

This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 10. It’s based mainly on counting and following the norms stated in the problem itself. Be careful while thinking !

ISI MStat PSB 2005 Problem 5 | Uniformity of Uniform

This is a simple and elegant sample problem from ISI MStat PSB 2005 Problem 5. It’s based the mixture of Discrete and Continuous Uniform Distribution, the simplicity in the problem actually fools us, and we miss subtle happenings. Be careful while thinking !

ISI MStat PSB 2012 Problem 2 | Dealing with Polynomials using Calculus

This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 2 based on calculus . Let’s give it a try !!

ISI MSTAT PSB 2011 Problem 4 | Digging deep into Multivariate Normal

This is an interesting problem which tests the student’s knowledge on how he visualizes the normal distribution in higher dimensions.

ISI MStat PSB 2012 Problem 5 | Application of Central Limit Theorem

This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 5 based on the Application of Central Limit Theorem.

ISI MStat PSB 2007 Problem 7 | Conditional Expectation

This is a very beautiful sample problem from ISI MStat PSB 2007 Problem 7. It’s a very simple problem, which very much rely on conditioning and if you don’t take it seriously, you will make thing complicated. Fun to think, go for it !!

ISI MStat Entrance Exam books based on Syllabus

Are you preparing for ISI MStat Entrance Exams? Here is the list of useful books for ISI MStat Entrance Exam based on the syllabus.

ISI MStat PSB 2008 Problem 8 | Bivariate Normal Distribution

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 8. It’s a very simple problem, based on bivariate normal distribution, which again teaches us that observing the right thing makes a seemingly laborious problem beautiful . Fun to think, go for it !!

ISI MStat PSB 2004 Problem 6 | Minimum Variance Unbiased Estimators

This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 6. It’s a very simple problem, and its simplicity is its beauty . Fun to think, go for it !!

ISI MStat PSB 2004 Problem 1 | Games and Probability

This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 1. Games are best ways to understand the the role of chances in life, solving these kind of problems always indulges me to think and think more on the uncertainties associated with the system. Think it over !!