My teacher gave me this tricky problem

please someone answer this question and write the solution as well.

the number of 6-digit numbers of the form ababab (in base 10) each of which is a product of exactly 6 distinct primes is (1) 8 (2) 10 (3)13 (4) 15 please reply friends.

Note by Abhigyan Adarsh
4 years, 2 months ago

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The 6digit6-digit numbernumber is in the form of ababab\color{#D61F06}{\overline{ababab}}.

Therefore ,

ababab\Rightarrow \color{#D61F06}{\overline{ababab}}

105a+104b+103a+102b+10a+b\Rightarrow 10^{5}a+10^{4}b+10^{3}a+10^{2}b+10a+b

101010a+10101b\Rightarrow 101010a+10101b

10101(10a+b) \Rightarrow 10101(10a+b)

3×7×13×37×(10a+b)\Rightarrow 3×7×13×37×(10a+b)

Now by observing the above line , we can say that (10a+b)(10a+b) should have two prime factors other than 3,7,133,7,13 and 3737.

Also when we see (10a+b)(10a+b) , we find that it is just a two digit number.

Now finding the number of numbers from 1010010-100 having two prime factors.

CaseICase I:

Taking 22 as one of the prime factors , we can say the upper limit as 2×50=1002×50=100 which means that the other prime number should be 2<P<502<P<50 as we discussed earlier that the number should be of 22 digits.

Therefore , number of prime numbers 2<P<502<P<50 is 1414 but we don't have to count 3,7,133,7,13 and 3737.Thus , there are 10\huge 10 such numbers.

CaseIICase II:

Now taking 55 as one of the prime factors , the upper limit is 5×20=1005×20=100. By this we can say that the another prime number should be 5<P<205<P<20.Now there are 55 prime numbers 5<P<205<P<20 but 77 and 1313 are not to be considered. Therefore , there are 3\huge 3 such numbers.

Only these are the possibilities.

Therefore , Answer=13\huge \boxed{13}

Akshat Sharda - 4 years, 2 months ago

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Nice!

Calvin Lin Staff - 4 years, 2 months ago

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Thanks!!\huge Thanks!!

Akshat Sharda - 4 years, 2 months ago

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Thanks Akshat for your amazing solution. You have proved the title to be wrong. I wish I could upvote your solution thousand times. And thank you Mohit for your answer.

abhigyan adarsh - 4 years, 2 months ago

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^_^

Akshat Sharda - 4 years, 2 months ago

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Answer is 13

Mohit Gupta - 4 years, 2 months ago

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you may be correct since i don't know the correct answer so please write the solution and thank you for your instant reply. i was not able to answer this question so i made a note.

abhigyan adarsh - 4 years, 2 months ago

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I think that you'll find my solution useful ^_^

Akshat Sharda - 4 years, 2 months ago

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Yes , its 1313.

Akshat Sharda - 4 years, 2 months ago

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