BdMO Problem 10

Number Theory Level 3

There are 2012 baskets in a row labeled as 1, 2, 3, ... 2012. A rabbit passes by and puts a carrot in every basket. The second rabbit does the same to each second basket; the third rabbit to each third basket and so on up to the 2012th rabbit. What is the total number of baskets with an odd number of carrots?

Image Credit: Flickr Tatiana .


The answer is 44.

Raiyun Razeen by Raiyun Razeen , 22, Bangladesh

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1 solution

Raiyun Razeen
Aug 13, 2015

We know, only perfect square numbers have an odd numbers of factors.

There are 44 perfect squares which are less than 2012.

So, there are 44 baskets with an odd number of carrots.

Answer 44 \boxed{44}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Don't we have to count prime numbers too?

Raghav Arora - 5 years, 8 months ago

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No! We can't count the prime numbers.

Think about a basket having a prime number in it's label.

Let it be 7 7

For the prime number 7 7 , we can write

7 = 1 × 7 \boxed {7 = 1 \times 7}

i.e. there will be 2 carrots in the 7th basket.

That's why we can't count the prime numbers .

Is it clear now? :)

Raiyun Razeen - 5 years, 7 months ago

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