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Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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14 Divisors

What is the smallest number with exactly 14 divisors?

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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?

Newspaper Sheets

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

If page $1$ is on the outside of the first sheet, then page $2$ is on the inside. Pages $3$ and $4$ are on the same sheet, as are pages $5$ and $6$. This therefore shares a sheet with page $7$, so is the inside of the sheet.

On the outside on the other side is therefore page $62$. The other three sheets have pages $63$ and $64$, $65$ and $66$ and $67$ and $68$ on them. Therefore there are $68$ pages, and four sheets per page, so $17$ sheets.


Alternatively, the sum of the page numbers on the same side of the sheet is always constant, since one side increases by $1$ every time the other decreases by $1$. This means the total is always $8 + 61 = 69$, so page $1$ shares with page $68$. Hence there are $68$ pages, so $17$ sheets.


 

This problem is taken from the UKMT Mathematical Challenges.

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