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Broad Topics >

Numbers and the Number System > Prime factors

##### Age 14 to 16 Challenge Level:

Lyndon chose this as one of his favourite problems. It is
accessible but needs some careful analysis of what is included and
what is not. A systematic approach is really helpful.

##### Age 11 to 14 Challenge Level:

Complete the following expressions so that each one gives a four
digit number as the product of two two digit numbers and uses the
digits 1 to 8 once and only once.

##### Age 14 to 16 Challenge Level:

Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.

##### Age 16 to 18 Challenge Level:

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5
What can you say about other solutions to this problem?

##### Age 11 to 14 Challenge Level:

The flow chart requires two numbers, M and N. Select several values
for M and try to establish what the flow chart does.

##### Age 16 to 18 Challenge Level:

Show that it is rare for a ratio of ratios to be rational.

##### Age 14 to 16 Challenge Level:

Take any pair of numbers, say 9 and 14. Take the larger number,
fourteen, and count up in 14s. Then divide each of those values by
the 9, and look at the remainders.

##### Age 14 to 16 Challenge Level:

Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)

##### Age 16 to 18 Challenge Level:

When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?

##### Age 16 to 18

An introduction to the ideas of public key cryptography using small
numbers to explain the process. In practice the numbers used are
too large to factorise in a reasonable time.

##### Age 16 to 18 Challenge Level:

How many divisors does factorial n (n!) have?

##### Age 11 to 14 Challenge Level:

Think of any three-digit number. Repeat the digits. The 6-digit
number that you end up with is divisible by 91. Is this a
coincidence?

##### Age 11 to 14 Challenge Level:

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides
exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest
power of two that divides exactly into 100!?

##### Age 11 to 14 Challenge Level:

Great Granddad is very proud of his telegram from the Queen
congratulating him on his hundredth birthday and he has friends who
are even older than he is... When was he born?

##### Age 14 to 16 Challenge Level:

In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?

##### Age 11 to 14 Challenge Level:

Find the highest power of 11 that will divide into 1000! exactly.

##### Age 11 to 14 Challenge Level:

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

##### Age 11 to 14 Challenge Level:

A number N is divisible by 10, 90, 98 and 882 but it is NOT
divisible by 50 or 270 or 686 or 1764. It is also known that N is a
factor of 9261000. What is N?

##### Age 14 to 16 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
power.

##### Age 16 to 18 Challenge Level:

The sum of the cubes of two numbers is 7163. What are these
numbers?