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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