### Consecutive Numbers

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

### 14 Divisors

What is the smallest number with exactly 14 divisors?

### 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?

# Star Sum

##### Stage: 3 Short Challenge Level:

The ten numbers in the star total 75.

Each number appears in two "lines".

And there are five lines, making a total of 150. Which implies the total for each line is 30.

This means that R+I=24 and therefore R=11 and I = 13 (or vice versa).

If R=11 then N=12, H =10 and C=11 which is impossible.

If R=13 then N = 10; H = 12 and C= 9, which is correct.

Alternatively:
once you know that each line adds up to 30 we can set up the following equations:
$(1) 4+N+R+3=30 ⇒N+R=23$
$(2) 1+R+I+5=30 ⇒R+I=24$
$(3) 3+I+C+7=30 ⇒I+C=20$
$(4) 4+H+C+5=30 ⇒H+C=21$
$(5) 1+N+H+7=30 ⇒N+H=22$
Then we are going to use a trick to “isolate R” by alternately adding and subtracting equations to get what we want, so consider
$(N+R)-(N+H)+(H+C)-(I+C)+(R+I)=23-22+21-20+24$
$⇒2R=26 ⇒R=13.$

This problem is taken from the UKMT Mathematical Challenges.
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