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Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

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The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

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Modular Knights

Try to move the knight to visit each square once and return to the starting point on this unusual chessboard.

Factorial Fun

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

The problem does not require much knowledge but it calls for careful reasoning and accurate use of standard notation. The problem provides scaffolding steps to help the problem solver think through the ideas needed to solve the problem.

Possible approach

Ask the class to count all the factors of $n!$ for $n = 2 , 3, 4$ and $5$ and suggest that they look for the most efficient way to do this. Then suggest that this problem might help them find the best way to do it.

This is not written in to the problem itself because EVERY time we try to solve a problem, unless it is very easy, we should think about first trying simple cases.

Key questions

How do we use the fact $24=2^3 \times 3$ to deduce that $24$ has $8$ factors? (Combinatorics question)
If we find the prime factors of $n$ how do we find out how many times these prime factors are repeated in $n!$ ?

Possible support

Try the problems Fac-finding, Powerful Factorial and Factoring Factorials. They are all special cases of this problem.

The problem Em'power'ed also focusses on equating the powers of prime factors.