
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 and itself. Now we know that is a -digit number. So it can be expressed in expanded form. The expanded form of is . Now we take all the 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 .
Complete step-by-step solution:
Now let us write in expanded form.
Upon doing so, we get the following :
.
Now let us all the common and see what we get.
Upon doing so, we get the following :
Now let us all the common and see what we get.
Upon doing so, we get the following :
Now let us all the common and see what we get.
Upon doing so, we get the following :
Now let us all the common and see what we get.
Upon doing so, we get the following :
Now must be a product of primes. It has to lie between since a number below would give me -digit number and a number above would give me a -digit number.
So now let us fix one prime number.
Let one prime number be . So the other prime number must be greater than . This would be our lower limit. We need to set our upper limit. Our upper limit would be since . So now, we should point out all the prime numbers between and excluding .That would be .
Now, let us fix another prime number.
Let the other prime number be . So the other prime number must be greater than . This would be our lower limit. We need to set our upper limit. Our upper limit would be since . So now, we should point out all the prime numbers between and excluding . That would be .
So the total numbers would be .
The number of -digit numbers of them form ( in base ) each of which is a product of exactly distinct primes is . So the answer is option C.
Note: Even if we try to fix prime numbers greater than as the lower limit excluding , 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.
Complete step-by-step solution:
Now let us write
Upon doing so, we get the following :
Now let us all the
Upon doing so, we get the following :
Now let us all the
Upon doing so, we get the following :
Now let us all the
Upon doing so, we get the following :
Now let us all the
Upon doing so, we get the following :
Now
So now let us fix one prime number.
Let one prime number be
Now, let us fix another prime number.
Let the other prime number be
So the total numbers would be
Note: Even if we try to fix prime numbers greater than
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