If a three digit positive integers ‘abc’ has 3 positive factors, how

Question

If a three digit positive integers ‘abc’ has 3 positive factors, how many positive factors does the 6-digit number ‘abcabc’ have?

A. 16 factors
B. 20 factors
C. 22 factors
D. 24 factors
E. 16 or 24 factors

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GMATinsight · Answer

Critical Observation : any number of the form 'abcabc' is divisible by 1001

and 1001 = 7*11*13

The Number abc has 3 factors which is possible only if it is of the form  a^2  i.e. a perfect square

Critical Observation : 3 digit perfect squares and square of a prime number

i.e. abc =  11^2, 13^2 , or a  newPrime^2

so abcabc = either  7*11^3*13  or  7*11*13^3  or  7*11*13*a^2  where a is a prime other than 7, 11 and 13

If  N = a^p*b^q*c^r... 
where a, b, c... are distinct primes
Number of factors of  N = (p+1)*(q+1)*(r+1)*...

Total Factors of  7*11^3*13  or  7*11*13^3  will be (1+1)*(3+1)*(1+1) = 16

Total Factors of  7*11*13*a^2  will be (1+1)*(1+1)*(1+1)*(2+1) = 24

Answer: Option E

To know the derivation of Property of ways of finding factors of a number is available in th video here

https://www.youtube.com/watch?v=uecuIrvaXGM

exc4libur · Answer

an integer with 3 factors is a prime squared
primes^2 that have three digits:
11,13,17,19,23,29,31

smallest: 11^2=121, abcabc=121121,
primef(abcabc)=11*11011=11*11*1001=11^ ,13^ ,7^ 
factors(abcabc)=( +1)( +1)( +1)=4*4=16 factors

largest: 31^2=961, abcabc=961961,
primef(abcabc)=31*31031=31*31*1001=31^2,11,13,7
factors(abcabc)=(3)(2)(2)(2)=3*8=24 factors

ans (E)

ScottTargetTestPrep · Answer

The numbers that have 3 positive factors are those that are a square of a prime. For example, 9 = 3^2 has three factors: 1, 3 and 9. Since here, abc has 3 factors, abc must be a square of a prime, i.e., abc = p^2 where p is a prime (note: The smallest value of p is 11 and the largest value of p is 31).

Since abcabc = abc x 1,001 and 1,001 = 7 x 11 x 13. If abc is neither 11^2 or 13^2, then abcabc = p^2 x 7^1 x 11^1 x 13^1 has (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 x 2 = 24 factors. If abc is 11^2, then abcabc = 11^2 x 7^1 x 11^1 x 13^1 = 7^1 x 11^3 x 13^1 has 2 x 4 x 2 = 16 factors. If abc is 13^2, then, like abc = 11^2, abcabc will also have 16 factors. Therefore, abcabc has either 16 or 24 factors.

Answer: E

Kinshook · Answer

Asked: If a three digit positive integers ‘abc’ has 3 positive factors, how many positive factors does the 6-digit number ‘abcabc’ have?

abcabc = 1000*abc + abc = 1001*abc
abcabc = 7*11*13*abc

Since abc has 3 factors, abc is of the form p^2 where p is a prime number

Case 1: p is either 7^2 or 11^2 or 13^2
Let us assume p = 7^2
abcabc = 7^3*11*13
Number of factors of abcabc = 4*2*2 = 16

Case 2: p is neither a multiple of 7 or 11 or 13
abcabc = 7*11*13*p^2
Number of factors of abcabc = 2*2*2*3 = 24

IMO E

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If a three digit positive integers ‘abc’ has 3 positive factors, how

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If a three digit positive integers ‘abc’ has 3 positive factors, how

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