What number between 50 and 60 has the most number of factors?

Answer 1

54 and 56 both have 8 factors .

60 has 12 factors. The others numbers all have fewer factors.

The answer will depend on the question you intended to ask. I suspect that the answer you want may be 60. Perhaps the following will prove instructive for you.

The question should be stated:

What number between 50 and 60 (inclusive) has the greatest number of factors?

OR

What number between 50 and 60 (inclusive) has the most factors?

OR

What number between 50 and 60 (exclusive) has the greatest number of factors?

OR

What number between 50 and 60 (exclusive) has the most factors?

Best of all is to ask: 'What integer, starting with 50 and ending with 60, has the greatest number of divisors?' - provided that this is what you meant to ask, of course.

All this may seem horribly pedantic to you! However, you will see, in a minute, why mathematicians insist on being so particular about nomenclature, in asking or answering mathematical questions.

In any case, I will assume that you mean the first or the second of the questions above. Here is the reasoning:

We can rule out the prime numbers, as they each have exactly two divisors: one and itself. This leaves 50, 51, 52, 54, 55, 56, 57, 58, and 60.

We can also rule out numbers that are the product of two primes, as they will have only four divisors; the first prime, the second prime, the number 1, and itself. Thus, we rule out 51, 55, 57, and 58. This leaves 50, 52, 54, 56, and 60.

Let's now discover a quick method of calculating the number of divisors of any given number. This will save a lot of trial-and-error and the corresponding likelihood of mistakes. We will use 56 as an example:

56 = 7 X 2 X 2 X 2 = 71 X 23. We have broken 56 down into its prime factors. Note the exponents, 1 and 3. Now, augment each exponent by one, and multiply: this gives 2 X 4 = 8. Eight is the number of divisors of 56. They may be shown as:

7 X 2 X 2 X 2 = 56;

7 X 2 X 2 = 28;

7 X 2 = 14;

7 X 1= 7;

2 X 2 X 2 = 8;

2 X 2 = 4;

2 X 1 = 2; and

1 = 1.

With the logic of this method now understood, we see that we can also rule out all numbers that are the product of a perfect square and a Prime number, as follows: The relevant exponents will be two and one; augmented and multiplied, they will give 3 X 2 = 6. Thus, we rule out 50 = 2 X 25 and 52 = 4 X 13, because they will have only six divisors. This leaves 54, 56, and 60 as the only possible candidates.

56, we have already done: it has eight divisors; by the same method, we see that

54 = 21 X 33 has 2 X 4 = 8 divisors; and

60 = 22 X 31 X 51 has 3 X 2 X 2 = 12 divisors. Aha!

We can now check our results, seeing that

60 = 1 X 60 = 2 x 30 = 3 X 20 = 4 X 15 = 5 X 12 = 6 X 10. Thus, the divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and, of course, 60 itself. It is instructive and also reinforcing to note that, of these twelve divisors, exactly six of them are less than 8. Eight is the first number whose square is larger than 60. Of the numbers less than 8, only one of them - 7 - does not divide evenly into 60. So, we must be on the right track.

Going back, now: PROVIDED that we meant to ask for the largest number of divisors belonging to a number from 50 to 60, INCLUSIVE, we find that 60 has the most divisors. If, on the other hand, we meant 'exclusive', rather than 'inclusive', (in which case, 50 and 60 don't count) then we find that 54 and 56 are tied for the most, at eight divisors each.

See, I wasn't being so horribly pedantic, after all! Just picky, out of necessity!
60, which has prime factors 2,3,5.