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Q: The n x indicating the number of factors of x?
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The n in X indicating the numbers of factors of X?

the exponent


The n in X indicating thenumber of factors of X?

nX in this case this says that n multiplys X by a factor of n Xn in this case this says that X is to the power of factor n not really another other case I can think of. Not sure what you are asking


Ten times the square of a non-zero number is equal to eighty times the number What is the number?

10 x n x n = 80 x n Divide by 10 x n; n = 8 Job done.


What is the smallest number with 20 divisors?

"We want to find the smallest number with exactly 20 factors (or divisors). If a number's prime factorisation consists of n different prime numbers and r repeated primes, then the total number of factors is 2n x (r+1)." * This is confusing. If the p.f. is 2r * p1 * ... * pn , where p1 to pn are n distinct odd primes, then yes. But if our number is N and factors as N = p1e1 . . . pkek , then N has this many factors (or divisors): d(N) = (e1 + 1)(e2 + 1) . . . (ek + 1). This is the general case. "The best solution to 2n x (r+1) = 20 is n = 2, r = 4." * This is better than n = 1, r = 9; N = 29 * 3 = 1536 > 240. "So the smallest number with exactly twenty factors, including itself, is 24 x 3 x 5 = 240. Its factors are 1,3,5,15,2,6,10,30,4,12,20,60,8,24,40,120,16,48,80,240." "The smallest number with exactly 20 factors excluding itself is, I think, 221 = 2097152." * This is wrong on two levels. You want a number with 21 divisors, including itself. First, 221 has 22 factors (21 not including itself), so you meant N = 220 = 1048576. Second, you can do much better by breaking up 21 = 7 * 3 so by the above formula (adding 1 to each exponent in the p.f.), use exponents of 6 and 2; to get the smallest N use 26 * 32 = 576, a mite smaller! * For more info see my page on supercomposite numbers at www.dansmath.com


What is the number with the most factors?

There isn't any, and it is quite simple to prove that. Suppose there is a number with the most factors and suppose that number is X. Now consider Y = 2*X. Y has all the factors of X and it has another factor, which is 2. So Y has more factors than X. This contradicts the statement that X has the most factors. Therefore, there is no number with the most factors.