Chung Kevin's favorite - (2)

Number Theory Level 1

$\begin{array}{c}(26 & = & 5+6+7+8 \\ 27 & = & 8+9+10 \\ 28 & = & 1+2+3+4+5+6+7 \\ 29 & = & 14+15 \\ 30 & = & 4+5+6+7+8 \\ 31 & = & 15+16 \end{array}$

If you see the above numbers , they can be represented as a sum of some consecutive numbers.If 32 can be expressed as $\displaystyle\sum_{i=1}^n a_i$ , where $a_1,a_2,a_3, \dots a_n$ are some consecutive positive numbers , compute $\displaystyle\prod_{i=1}^n a_i$ .

Note: If you think that 32 is not lucky enough to have such representation , input the answer as 999.

Try more by clicking here .

The answer is 999.

2 solutions

Sravanth C.
May 30, 2015

Actually Nihar was taking of polite numbers.

In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. Other positive integers are impolite

The first few polite numbers are,

$3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, \\ 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, \\ 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, \\ 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, . . . . . .$

The impolite numbers are exactly the powers of two.It follows from the Lambek–Moser theorem that the nth polite number is $ƒ(n + 1)$ , where

$f(n)=n+\left\lfloor\log_2\left(n+\log_2 n\right)\right\rfloor.$

For more information follow this link

All Content taken from wikipedia, this is for the sole purpose of education, it is in no way intended for copyright violations.

Can you prove why all numbers of the form $2^n$ cannot have such representation.?

Nihar Mahajan - 6 years ago

Sorry, as I told you ealier, that I am confined to a small screen I can't type the solutions and currently I am travelling, if you want to know more read this .

Also please don't mind that I didn't solve most of you JEE Novices, because I am out of station. . . . $\huge\ddot\frown$

Sorry for a late reply, the signal is not proper. . . . .

Sravanth C. - 6 years ago

Learnt a bit of politeness today.

Kaushik Chandra - 3 years, 6 months ago

Daniel Ryan
May 30, 2015

You might want to specify these numbers are positive integers. The sum of the numbers from -31 to 32 consecutively equals 32 (product being zero).

Thanks for telling that. Also please refrain from writing reports in solution section.You can submit the report by clicking the "dot-dot-dot" menu in the right bottom part of the screen. Thanks!

Nihar Mahajan - 6 years ago

My apologies. It was my first time making a comment on this site.

Daniel Ryan - 6 years ago

Well , it was an advice and not a scolding. So no need to apologize.

Nihar Mahajan - 6 years ago

Chung Kevin's favorite - (2)

Try more by clicking here .

The answer is 999.

2 solutions

All Content taken from wikipedia, this is for the sole purpose of education, it is in no way intended for copyright violations.

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