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

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Joined: 02 Sep 2009
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If a three digit positive integers ‘abc’ has 3 positive factors, how  [#permalink]

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14 Apr 2020, 03:27
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41% (01:32) correct 59% (02:37) wrong based on 37 sessions

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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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If a three digit positive integers ‘abc’ has 3 positive factors, how  [#permalink]

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14 Apr 2020, 03:39
Bunuel wrote:
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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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

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

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Re: If a three digit positive integers ‘abc’ has 3 positive factors, how  [#permalink]

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26 May 2020, 10:44
Bunuel wrote:
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

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^[3],13^[1],7^[1]
factors(abcabc)=([3]+1)([1]+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

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Re: If a three digit positive integers ‘abc’ has 3 positive factors, how  [#permalink]

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30 May 2020, 13:30
Bunuel wrote:
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

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.

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Re: If a three digit positive integers ‘abc’ has 3 positive factors, how   [#permalink] 30 May 2020, 13:30