Question

What is the maximum possible length of divisor chain of 600

Answer 1

Answer:

24

Steps:

Method 1:

If x=am∗bn∗cp...x=am∗bn∗cp...

where a, b, c, ... are the prime factors of x, then the number of factors of x is

(m+1)(n+1)(p+1)...

This is because any factor can be made by selecting 0 to m numbers of a (in m+1 ways), 0 to n numbers of b (in n+1 ways) and so on.

Since 600=23∗3∗52600=23∗3∗52

Total number of factors of 600 = (3+1)(1+1)(2+1) = 24

Method 2:

600−−−√≈24.5600≈24.5

Factors of 600 which are less than 600−−−√600 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24

These are 12 factors.

Corresponding to each of these factors there will be exactly one factor >600−−−√>600.

Hence total number of factors = 12*2 = 24

Answer 2

Answer:

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