Select Page

Understand the problem

Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r>1$ ,are of the form $n=2^l$ for some $l\ge 0$.

Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2019. Subjective Problem no. 1.

Number Theory

Difficulty Level

8.5 out of 10

Do you really need a hint? Try it first!

Claim: Any positive integer $n$ can be written as $n=2^k\cdot m$ , where $k\ge0$ and $m$ is an odd positive integer. To prove this claim use the fact :  $n=2^k\cdot p_1^{k_1}\cdot p_2^{k_2}\cdots p_i^{k_i}$, where $k$ and all $k_i$ are non-negetive integers and all $p_i$ are odd primes.

The sum of (any) $r$ consecutive positive integers is given by, $(q+1)+(q+2)+(q+3)+\cdots+(q+r)$ =   $q\cdot r+(1+2+3+\cdots+r)$ =   $q\cdot r+\frac{r(r+1)}{2}$.

Equating this sum to $n$ we get,           $2^k\cdot m = q\cdot r+\frac{r(r+1)}{2}$   Or,     $2^{k+1}\cdot m = 2q\cdot r+r(r+1)$   Or,     $2^l\cdot m = r(2q+r+1)$ , where $l=k+1\ge1$.

In particular if we take $n=2^l$ then $m$ is equal to 1. Since both $r$ and $(2q+r+1)$ are greater than 1, so they can’t be equal to $m$ in this case. Again one of $r, (2q+r+1)$ is odd integer which implies the product $r(2q+r+1)$ can’t be equal to $2^l$.  $\Rightarrow 2^l \neq r(2q+r+1)$ $\Rightarrow 2^l$ can’t be expressed as the sum of $r$ consecutive positive integers with $r>1$ and $l\ge 1$. Now, $n=2^0=1$ also can’t be written in the same manner when $l=0$. Therefore the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r>1$ , are of the form $n=2^l$ for some $l\ge 0$. To show that any number other than of the form $2^l$ is sum of consecutive integers:
Observe that for $n=2^l.m$, where $m$ is an odd number greater than 1, there are two cases:
1. $m < 2^l$
Select $r$ and $q$ such that $r = m$ and $2q+r+1 = 2^l$. 2. $m > 2^l$
Select $r$ and $q$ such that $2q+r+1 = m$ and $r = 2^l$. (QED).

I.S.I. & C.M.I. Entrance 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 !!