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The number of 6-digit numbers of them form ababab (in base 10) each of which is a product of exactly 6 distinct primes ?

Answer
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Hint: This is a question from number theory. We have to solve them mostly through comparison. Prime numbers are those numbers whose factors are 1 and itself. Now we know that ababab is a 6-digit number. So it can be expressed in expanded form. The expanded form of ababab is 105a+104b+103a+102b+101a+b . Now we take all the 10s common and see what we get . Upon doing so, we write the number we get as a product of primes. After doing that, we draw out conditions for the remaining relation between a,b.

Complete step-by-step solution:
Now let us write ababab in expanded form.
Upon doing so, we get the following :
ababab=105a+104b+103a+102b+101a+b .
Now let us all the a common and see what we get.
Upon doing so, we get the following :
ababab=105a+104b+103a+102b+101a+bababab=a(105+103+101)+104b+102b+b
Now let us all the b common and see what we get.
Upon doing so, we get the following :
ababab=105a+104b+103a+102b+101a+bababab=b(104+102+1)+a(105+103+101)
Now let us all the 10s common and see what we get.
Upon doing so, we get the following :
ababab=105a+104b+103a+102b+101a+bababab=b(104+102+1)+a(105+103+101)ababab=b(104+102+1)+10a(104+102+1)
Now let us all the 104+102+1 common and see what we get.
Upon doing so, we get the following :
ababab=105a+104b+103a+102b+101a+bababab=b(104+102+1)+a(105+103+101)ababab=b(104+102+1)+10a(104+102+1)ababab=104+102+1(10a+b)ababab=10101(10a+b)ababab=3×7×13×37(10a+b)
Now 10a+b must be a product of primes. It has to lie between 10100 since a number below 10 would give me 5-digit number and a number above 100 would give me a 7-digit number.
So now let us fix one prime number.
Let one prime number be 2. So the other prime number must be greater than 2. This would be our lower limit. We need to set our upper limit. Our upper limit would be 50 since 2×50=100. So now, we should point out all the prime numbers between 2 and 50 excluding 3,7,13,37.That would be 10.
Now, let us fix another prime number.
Let the other prime number be 5. So the other prime number must be greater than 5. This would be our lower limit. We need to set our upper limit. Our upper limit would be 20 since 5×20=100. So now, we should point out all the prime numbers between 5 and 20 excluding 3,7,13,37. That would be 3.
So the total numbers would be 10+3=13.
The number of 6-digit numbers of them form ababab( in base 10) each of which is a product of exactly 6 distinct primes is 13. So the answer is option C.

Note: Even if we try to fix prime numbers greater than 5 as the lower limit excluding 3,7,13,37, we would get the same combination number again. So this would be a repetition. We would get more numbers than there are. Questions from Number theory appear pretty simple but the concepts involved in the solution are a little complex. To build up the intuition for every solution, we should be exposed to a lot of problems. So we should practice thoroughly.

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