Counting Factors

Is there an efficient way to work out how many factors a large number has?

Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Elevenses

Stage: 3 Challenge Level:

Megan from the Thomas Deacon Academy used a spreadsheet to find all 28 pairs of numbers that add up to a multiple of 11. This is how she did it:

To work out the first box I started with 9, adding the rest of the numbers, and then moved on to 46, but didn't do 46 add 9 since it had already been done. Ithen carried this out through out the table but making sure I had not done any in front of the number I was working on as it would have already been done.

Then I highlighted all the answers that are in the 11 times table.
In the attached document are my notes.

Alex from St. Anne's School noticed something special about the numbers in some of the pairs:

There were over 25 different pairs of numbers wich totalled a multiple of 11.
We noticed that the numbers we added to 9, 20 and 31 were all the same:

9+46=55
9+79=88
9+13=22
9+90=99
9+2=11

20+46=66
20+79=99
20+13=33
20+90=110
20+2=22

31+46=77
31+79=110
31+13=44
31+90=121
31+2=33

The difference between 9 and 20 is 11 and the difference between 20 and 31 is 11.
When we added 11 to 31 and made 42. We added 46, 79, 13, 90 and 2 to this number and found that each result was a multiple of 11.

Jack and Zaim from London sent us this very clear explanation of why this happens.

Curtis from Shatin College used a similar strategy :

I divided all of the numbers by 11 and wrote down their remainders. Then I wrote a chart of them, in the same spot. After that, I checked in the remainder box for any pairs that added up to 11. Finally I transfered the numbers back on to the provided grid, and came up with 28 solutions for both the 11's and the 13's.

Adil from Valentines High School discovered the same property of the numbers that could be paired:

We made a spreadsheet to add all the pairs in the grid and we found a rule:
• The numbers that are of the form 11X+2 or 11X-2 will pair up to give a multiple of 11.
• Obviously pairs of multiples of 11 will add to be a multiple of 11.
• Finally single digit numbers will pair with another number of the form 11X minus the single digit number itself.