What is the greatest number of factors that a positive ineger less than 100 can have?

Question

help

Answer 1

The question did not restrict factors to prime numbers. A factor is any divisor that leaves no remainder:


From Wikipedia:

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder.

If you include 1, 60 has 12; factors: 1,2,3,4,5,6,12,15,20,30,60
72 has 12 factors: 1,2,3,4,6,8,9,12,18,24,36,72
96 also has 12: 1,2,3,4,6,8,12,16,24,32,48,96

I suspect 12 is the max. Note that all these numbers have 12 as a factor, and 12 has 6 factors.

Answer 2

100 can be broken down into 10 times 10
so every number less than 100 will have its largest factor no greater than 10
all factors are prime
we need to see how many factors are 10 or less
the primes below 10 are 2, 3, 5, 7
so, the most factors a number below 100 can have is these 4

Answer 3

Both 60 and 72 have 12 factors... not too sure if this is the maximum
Basically take a number and decompose into prime factors
i.e n= p^a * q^b * r^c and so on..
so number of factors = (a+1) * (b+1) * (c+1)
eg. 60 = 2^2 * 3 ^1 * 5^1
so no of factors = 3 * 2 * 2 =12

Hope this helps