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If a three digit number 'abc' has 3 factors, how many factors does the 6 -digit number 'abcabc' have?
- a)16 factors
- b)24 factors
- c)16 or 24 factors
- d)20 factors
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If a three digit number abc has 3 factors, how many factors does the 6...
Explanation:
Factors of a Three-Digit Number:
- For a three-digit number abc to have 3 factors, it must be a perfect square.
- The only perfect squares between 100 and 999 are the squares of numbers 10 to 31.
- These numbers are 100, 121, 144, 169, 196, 225, 256, 289, and 324.
- So, the three-digit numbers with 3 factors are 100, 121, 144, 169, 196, 225, 256, 289, and 324.
Factors of a Six-Digit Number:
- For the 6-digit number abcabc to be formed, it must be a multiple of a three-digit number with 3 factors.
- The possible three-digit numbers with 3 factors are 100, 121, 144, 169, 196, 225, 256, 289, and 324.
- Therefore, the 6-digit number abcabc can be formed by repeating these three-digit numbers.
- Each of these three-digit numbers has a certain number of factors, and when repeated twice to form a 6-digit number, the total factors will be a combination of the factors of the original three-digit number.
- The number of factors for each of these three-digit numbers is 16, except for 144 which has 15 factors.
- So, the 6-digit number abcabc can have either 16 factors (if the 3-digit number is 144) or 24 factors (for all other three-digit numbers).
Therefore, the 6-digit number abcabc can have either 16 factors or 24 factors. Hence, the correct answer is option 'C'.
Factors of a Three-Digit Number:
- For a three-digit number abc to have 3 factors, it must be a perfect square.
- The only perfect squares between 100 and 999 are the squares of numbers 10 to 31.
- These numbers are 100, 121, 144, 169, 196, 225, 256, 289, and 324.
- So, the three-digit numbers with 3 factors are 100, 121, 144, 169, 196, 225, 256, 289, and 324.
Factors of a Six-Digit Number:
- For the 6-digit number abcabc to be formed, it must be a multiple of a three-digit number with 3 factors.
- The possible three-digit numbers with 3 factors are 100, 121, 144, 169, 196, 225, 256, 289, and 324.
- Therefore, the 6-digit number abcabc can be formed by repeating these three-digit numbers.
- Each of these three-digit numbers has a certain number of factors, and when repeated twice to form a 6-digit number, the total factors will be a combination of the factors of the original three-digit number.
- The number of factors for each of these three-digit numbers is 16, except for 144 which has 15 factors.
- So, the 6-digit number abcabc can have either 16 factors (if the 3-digit number is 144) or 24 factors (for all other three-digit numbers).
Therefore, the 6-digit number abcabc can have either 16 factors or 24 factors. Hence, the correct answer is option 'C'.
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Community Answer
If a three digit number abc has 3 factors, how many factors does the 6...
'abc' has exactly 3 factors, so 'abc' should be square of a prime number. (This is an important
inference, please remember this).
Any number of the form paqbrC will have (a+1)(b+1)(c+1) factors, where p,q,r are prime.
So if a number has 3 factors, its prime factorization has to be p2.
'abcabc′=′abc′×1001 or abc×7×11×13 (again, this is a critical idea to remember)
Now, 'abc' has to be square of a prime number. It can be either 121 or 169 (square of either 11 or 13 ) or it can be the square of some other prime number.
When abc=121 or 169, then 'abcabc' is of the form p3q1r11, which should have 4×2×2=16 factors.
When 'abc' = square of any other prime number (say 172 which is 289) , then 'abcabc' is of the
form p1q1r1s2, which should have 2×2×2×3=24 factors
So, 'abcabc' will have either 16 factors or 24 factors.
inference, please remember this).
Any number of the form paqbrC will have (a+1)(b+1)(c+1) factors, where p,q,r are prime.
So if a number has 3 factors, its prime factorization has to be p2.
'abcabc′=′abc′×1001 or abc×7×11×13 (again, this is a critical idea to remember)
Now, 'abc' has to be square of a prime number. It can be either 121 or 169 (square of either 11 or 13 ) or it can be the square of some other prime number.
When abc=121 or 169, then 'abcabc' is of the form p3q1r11, which should have 4×2×2=16 factors.
When 'abc' = square of any other prime number (say 172 which is 289) , then 'abcabc' is of the
form p1q1r1s2, which should have 2×2×2×3=24 factors
So, 'abcabc' will have either 16 factors or 24 factors.
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If a three digit number abc has 3 factors, how many factors does the 6 -digit number abcabc have?a)16 factorsb)24 factorsc)16 or 24 factorsd)20 factorsCorrect answer is option 'C'. Can you explain this answer?
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