How many ways are there of writing XXXX as a sum of positive consecutive integers?

Interesting Yahoo! Answers thread.

I found this discussion very interesting (after coming upon it after a search due to encountering one of these problems on FTW and failing horribly).

Basically, the result is that the number of ways of writing a number as a sum of consecutive integers is the number of odd divisors of that number, minus one. This is an awesome conclusion.

How many ways are there of writing 123123 as a sum of positive consecutive integers?

Therefore, the number of factors (all of which must be odd, because there is no factor of in ) are:

Since there are 32 odd factors of 123123, the number 123123 can be written as a sum of positive consecutive integers in 31 ways.

Each of these ways is one of the odd divisors of 123123 (we subtract 1 because we can’t count the divisor 1 as that would yield only 1 integer, 123123 itself, and we need plural integers anyways).

If we count the even divisors, that will result in some of the divisors being negative. (See thread.)

Ok, lame post, sorry. Guys, do some of the earlier problems, like that staircase one ^_^

Lol, just saw this on the math team forum, was posted by like last last year, but anyways…

In the prime factorization of the expression there is a term. Find x.

» Read more: Variation of a previous number theory problem

1) Written out in base 10, the number ends in how many zeroes?

Each zero in a number represents a factor of 10, or a factor each of five and two.

Therefore, to find the number of zeros after a number, we need to find the number of factors of five and the number of factors of two in .

How many factors of five are in ?

Simply divide 345 by five.

However, we need to count the factors of five squared (25) twice, so we divide again by 25.

And finally, let’s not forget the factors of five cubed which need to be counted three times.

Summing these together:

Thus there are 84 zeroes after the number .

2) The product of the 12-digit number and the 12-digit number is the 24-digit number . Find the digit .

Did you try to find the digits that affected the value of A and try multiplying things out?

A smarter solution, do this calculation mod 9. It’s called “casting out nines”. Cross out all the nines and all the numbers that add up to nine, etc, and add up the remaining numbers, subtracting nine when needed.

Therefore,

And

The product mod 9 is:

And since the products are supposed to be equal, they will also be equal mod 9.

It’s as simple as that!