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.

Topic

Number Theory

Difficulty Level

8.5 out of 10

Suggested Book

Elementary Number Theory by David M. Burton

Start with hints

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).

Connected Program at Cheenta

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.

Similar Problem

An inequality with many unknowns

Understand the problemLet be positive real numbers such that . Prove thatSingapore Team Selection Test 2008InequalitiesMediumInequalities by BJ VenkatachalaStart with hintsDo you really need a hint? Try it first!Use the method of contradiction.Suppose that $latex...

Looks can be deceiving

Understand the problemFind all non-zero real numbers which satisfy the system of equations:Indian National Mathematical Olympiad 2010AlgebraMediumAn Excursion in MathematicsStart with hintsDo you really need a hint? Try it first!When a polynomial equation looks...

IMO, 2019 Problem 1 – Cauchyish Functional Equation

This problem is a patient and intricate and simple application of Functional Equation with beautiful equations to be played aroun with.

A sequence of natural numbers and a recurrence relation

Understand the problemDefine a sequence by , andfor For every and prove that divides. Suppose divides for some natural numbers and . Prove that divides Indian National Mathematical Olympiad 2010 Number Theory Medium Problem Solving Strategies by Arthur Engel...

Linear recurrences

Linear difference equationsA linear difference equation is a recurrence relation of the form $latex y_{t+n}=a_1y_{t+n-1}+a_2y_{t+n-2}+\cdots +a_ny_t+b$. If $latex b=0$, then it is called homogeneous. In this article, we shall also assume $latex t=0$ for...

2013 AMC 10B – Problem 5 Maximizing the Difference:

This is based on simple ineqaulities on real numbers.

An inductive inequality

Understand the problemGiven and for all , show that Singapore Mathematical Olympiad 2010 Inequalities Easy Inequalities by BJ Venkatachala Start with hintsDo you really need a hint? Try it first!Use induction. Given the inequality for $latex n=k$, the inequality...

A search for perfect squares

Understand the problemDetermine all pairs of positive integers for which is a perfect square.Indian National Mathematical Olympiad 1992 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!First consider $latex n=0$....

INMO 1996 Problem 1

Understand the problema) Given any positive integer , show that there exist distint positive integers and such that divides for ; b) If for some positive integers and , divides for all positive integers , prove that .Indian National Mathematical Olympiad...

Trigonometric substitution

Understand the problemLet with . Prove thatDetermine when equality holds.Singapore Team Selection Test 2004InequalitiesMediumInequalities by BJ VenkatachalaStart with hintsDo you really need a hint? Try it first!Show that there exists a triangle $latex \Delta ABC$...