a natural number has exactly 10 divisors including 1 and itself. how many distinct prime factors can this natural number can have?

• This mathematical problem can be explained well if we use a basic formula which helps to find out the the No. of Divisors or Factors of a particular Numbers it has, it generally goes like this.
  
  If N is a Natural number, Prime Factorization of N = a^x * b^y .
  Then the No. of divisors = (x+1)*(y+1)
  
  As the Natural number has exactly 10 divisors & 10 = 5 * 2
  So possible value of x = 4 & y = 1 or x = 9 .
  --------------------------------------------------------------------------------
  Example
  if 512 = 2^9 , total No. of factor = (x+1) i.e (9+1) = 10
  or
  if 48 = 2^4*3^1 , total No. of factor = (x+1)*(y+1) i.e (4+1)*(1+1) = 10
  or
  if 19683 = 3^9 , total No. of factor = (x+1) i.e (9+1) = 10
  or
  if 162 = 2^1*3^4, total No. of factor = (x+1)*(y+1) i.e (4+1)*(1+1) = 10
  --------------------------------------------------------------------------------
  Thus Answer = Either 1 or 2 distinct prime factors.
  
• 9 years agoHelpfull: Yes(70) No(3)
• ans is either 1 or 2 .
  example if the no is 512 . then the divisors of this no. are
  1,2,4,8,16,32,64,128,256,512;
  
  so only one prime factors are there 2;
  
  if the no is 48 the divisors are
  1,2,3,4,6,8,12,16,24,48;
  so the prime factors are 2,3;
• 11 years agoHelpfull: Yes(49) No(8)
• for eg:48 is the number with 10 divisors 1,2,3,4,6,8,12,16,24,48 and hence only 2 are possible i.e 2 and 3
• 11 years agoHelpfull: Yes(19) No(9)
• 2 prime factors are possible only in case of natural number having exactly 10 divisors including 1 and itself.
• 12 years agoHelpfull: Yes(10) No(11)
• it will hav 2 prime factor
  as 10=5*2 thus no. will be of form (prime no.1)^4 * (prime no.2)
  for eg 48= divisors(1 2 4 8 16 3 6 12 24 4 48)= 2^4*3
  (it consist of two prime no.)
• 10 years agoHelpfull: Yes(7) No(2)
• There is only two distinct prime factors.
  If No. of divisors are 10, then the expression of prime factors would be:
  
  P^a+Q^b and then the no. of factors would be (a+1)(b+1) which 10 is possible.
  If there are more than two prime factrors, 10 is not possible.
• 12 years agoHelpfull: Yes(6) No(14)
• The number will be a multiple of 8 prime factors. Then it will be divisible by 1 and the number itself i.e it will have exactly 10 divisors.
  
  Example:
  
  3*7*11*13*17*19*23*29 = 646969323
  This number has exactly 10 divisors including 1 and itself.
• 12 years agoHelpfull: Yes(3) No(43)
• as we 9 no. of factors=(p+1)(q+1).... now see if no. is n ,then divisors may be 2,3,5,or 7 or other 2 digit prime nos. . as 4 ,6,8,9 or other 2 digit nos. can be obtained by using above digits for example 4=2^2,9=3^2 nd so on.thus no n =a^p+b^q+c^r....and given no. of factor=10.so (p+1)(q+1)(r+1)....=10.as 10 can be broken down as 2*5 or 10*1, or vice versa.in each case for example 2*5=we can use at most 2 digits(2 digits m using as we r talking abt powers to a number) as either p+1=2 & q+1=5, or p+1=10 &q+1=1.as p being value of power .each of them have atmost 2 values i.e 2 when 2*5 and 1 when 10*1 (as in this case q=0 and p & q are powers so if power goes o no. goes to 1)so 1 when a^9 and 2 when a^4*b^1(for the guys who don't 9 no of factors rule no of factors of a no. n which can be represented as a^p*b^q,means no. can be written as power of prime no.s a and b is always equal to p+1*q+1.)
• 10 years agoHelpfull: Yes(3) No(1)
• as if no is 512 then
  we can say that divisors are 2^9
  so it will have only one prime divisor which is 2
• 9 years agoHelpfull: Yes(2) No(2)
• 2 IS THE ANSWER
  10 FACTORS =(1+1)(4+1)
  EXAMPLE 2*3*3*3*3
  OR 3*5*5*5*5
  .............
  MANY MORE BUT ALL THE NUMBERS WILL HAVE ONLY 2 DISTINCT
  PRIME FACTORS
• 10 years agoHelpfull: Yes(1) No(0)
• there can be atmost 8 prime factors
  for eg 30 = 2*3*5 -3 factors along with 1 and 30 itself
  similarly 210 = 2*3*5*7 - 4 factors
  similarly if question says there are 10 factors including 1 an self then we can multiply upto 8 consecutive prime no.
• 9 years agoHelpfull: Yes(1) No(0)
• I think 4...
  
• 9 years agoHelpfull: Yes(1) No(1)
• A number can be written in form of (p^a)(q^b)(r^c) where p,q,r are its prime factors.
  then number of factors = (a+1)(b+1)(c+1)
  in case of 10 factors, it can be :
  1*10 or 2*5
  i.e.
  (p^9) ---> factors = (9+1) = 10 or,
  (p^1)(q^4) --> factors = (1+1)(4+1) = 10
  so, in first case no. of prime factors = 1 (i.e. p)
  and in second case number of prime factors = 2 (i.e. p,q)
  Ans : either 1 or 2 distinct prime factors.
• 5 years agoHelpfull: Yes(1) No(0)
• ans is 1 or 2
  as --- given 10 including 1 and itself means actually it has 8 divisors
  means 8=4*2
  or 8= 8*1
  thats why
• 8 years agoHelpfull: Yes(0) No(0)
• maximum 9 prime factor can have ....
• 7 years agoHelpfull: Yes(0) No(0)