To work out the number of factors 360 has, I would adopt the same method Charlie uses of prime factorisation to give 360=2^3×3^2×5 and from here I would gather the exponents or indices which are 3,2 and 1 add one to each one respectively to give 4,3,2 and multiply them together: 4*3*2=24 therefore the number 360 has 24 factors. This is a very efficient and convenient method for calculating the number of factors a certain number has post prime factorisation of that number.
Charlie and Alison think all of these numbers have exactly 24 factors. Can you see why?
1) 25725=5^2×3^1×7^3
2) 217503=11^1×13^3×3^2
3) 312500=5^7×2^2
4) 690625=17^1×13^1×5^5
5) 94143178827=3^23
I would implement the same strategy I used to figure out the number of factors 360 through multiplying the exponents together after increasing each exponent by one: Every integer N is the product of powers of prime numbers:
N = p^α+qβ· ... · r^γ.
Where, p, q, ..., r are prime, while α, β, ..., γ are positive integers. If N is a power of a prime, N = p^α, then it has α + 1 factors:
Using this method:
1) (2+1)*(1+1)*(3+1)= 3*2*4=24 factors
2) (1+1)*(3+1)*(2+1)=2*4*3=24 factors
3) (7+1)*(2+1)=8*3=24 factors
4) (1+1)*(1+1)*(5+1)= 2*2*6=24 factors
5) (23+1)= 24 factors
To answer these questions -
How can I find a number with exactly 14 factors?
How can I find the smallest such number?
How can I find a number with exactly 15 factors?
How can I find the smallest such number?
How can I find a number with exactly 18 factors?
How can I find the smallest such number?
Which numbers have an odd number of factors?
The formula (α + 1)(β + 1) ... (γ + 1) can greatly help. For example, to answer the first question, i can recognise that 14=2*7 therefore meaning that the number being decomposed has only two prime factors. In order to make n as small as possible, I have to choose the two smallest primes being 2 and 3 and so: A number with exactly 14 divisors can be of one of the two following forms , either p^6 * q
OR
p^13
where p and q are prime.
From here it should be obvious that the smallest such number is 2^6 * 3 = 192
Smallest possible number with exactly 15 divisors: Since 15 = (3)(5) any number with 15 divisors would have to factor into either
p^14 or (p^2)(q^4)
The smallest number of the form p^14 is 2^14 = 16,384
The smallest number of the form (p^2)(q^4) = (3^2)(2^4) = (9)(16) = 144
Therefore, the answer would have to be 144
Smallest number with exactly 18 factors: 18= 18*1 or 9*2 or 6*3 or 2*3*3=2*3^2
The first three prime numbers listed above are 2,3 and 5.Increasing the exponents of 2 and 3 gives me 2^2*3^2*5=180.
The numbers which have an odd number of factors are always square numbers because you have to include the repeated factor of that number which is its square root. For example, Factors of 36 are
1 and 36
2 and 18
3 and 12
4 and 9
6 and 6
The repeated factor of 6 will be omitted if we list the factors hence why 36 has 9 factors which is odd.
Extension:
What is the smallest number with exactly 100 factors?
If n=100. We can write it as
100=5×5×2×2100=5×5×2×2
Relative to this we have the number 24×34×5×7=4536024×34×5×7=45360. This is the smallest number with 100 factors.
Which number less than 1000 has the most factors?
840 with 32 factors even though it is a composite number rather than a prime number.