Factor This 10-Digit Number

Note that $(10^{x}+1)^{3}=10^{3x} + 3 \times 10^{2x} + 3 \times 10^{x} + 1$

For x=1, $11^{3}=1331$

For x=2, $101^{3}=1030301$

And for x=3, $1001^{3}=1003003001$

By quick check of factoring rules, we see 1001 is divisible by 7, 11, and therefore 13, so $1003003001=7^{3} \times 11^{3} \times 13^{3}$ which gives us $\boxed{9}$ prime factors.

We notice the symmetry in the number. 1,3,3,1. This should remind us of Pascal's triangle. In particular, the expansion $(x+1)^3=x^3+3x^2+3x+1$ . Letting $x=10^3$ we get $(10^3+1)^3=10^9+3\cdot 10^6+3\cdot 10^3+1=1003003001$ , which is our number. So we want the number of prime divisors of $(10^3+1)^3=(1001)^3=(7\cdot 11\cdot 13)^3=7^3\cdot 11^3\cdot 13^3$ , so the answer is $3+3+3=\boxed{9}$ .

Remark: 1001=7 11 13 is a very common factorization in the math competition world - strap it to your tool belt!

1003003001 = 7^3 * 11^3 * 13^3 ,so 1003003001 has a 3 + 3 + 3 = 9 product

$1003003001$ is $1001^3$ because $(1000+1)^3 = 1000^3+3\cdot1000^2+3\cdot1000+1$ $1001 = 7\cdot11\cdot13$ , 3 primes. When you cube it, you multiply these three times, to get 9 primes.

1003003001= (1001)^3 = (7 × 11 × 13)^3 = 7^3 × 11^3 × 13^3

total primes = 3 + 3 + 3 = 9

Prime factorisation of 1003003001 is

                   7 X 7 X 7 X 11 X 11 X 11 X 13 X 13 X 13

They are 9.

The number 1003003001=7x7x7x11x11x11x13x13x13 is the product of 9 primes.

1 003 003 001 = 1001 x 1001 x 1001 = (7 x 11 x 13)^3 3 x 3 = 9

From the binomial theorem (or Pascal's triangle) we see that 1003003001 is a perfect cube

∛(1003003001) = 1001 = 7•11•13, which has 3 prime factors, so N = 3•3 = 9

when we do prime factorization of 1003003001 we get, 11^3 * 13^3 * 7^3 so there are total 9 prime factors....:)

A simple method for ESTIMATING the number, n, of prime factors for a large number N can be used:

n ~ ln ( ln ( N ) ) ^ 2

For 1003003001, ln ( ln ( 1003003001 ) ) ^ 2 = 9.1894... ~ 9

This estimation becomes more and more accurate with greater numbers

Very Simple it's 7 x 7 x 7 x 11 x 11 x 11 x 13 x 13 x 13 =1003003001 All are prime and total are 9!!

Start by dividing with prime no. As u will go u will see it will result in.. 7x7x7x11x11x11x13x13x13

1003003001 = 1001^3 1001= 1 7 11*13 total 3 prime numbers, 3 times. That makes it 9

Complete proof :

Note that $1003003001 = 7^3 \cdot 11^3 \cdot 13^3$ and $7,11,13$ are primes, so it is the product of $N = 3+3+3 = \boxed{9}$ primes.

Motivation :

$1003003001$ looks similar to $1331 = 11^3$ . Indeed, if we remember the binomial expansion of $(x+1)^3 = x^3 + 3x^2 + 3x + 1$ , we can see that $11^3 = 1331$ can be obtained from the binomial expansion by $x=11$ . In this case, $1003003001$ can be obtained from the binomial expansion by $x=1000$ , so $1003003001 = 1001^3$ . Finally, $1001 = 7 \cdot 11 \cdot 13$ is a well known factorization.

WE NEED TO PRIME FACTORIZE THE GIVEN NUMBER. 1003003001= 7 x 7 x 7 x 13 x 13 x 13 x 11x 11 x 11

Test first for the factors by dividing 1003003001 to all possible prime numbers, starting from 1, 2, 3, 5, 7, 11, 13, so on.. and check which will result a whole number.

Then you'll notice that it divides by 7, 11, and 13, resulting into whole number.

Prime factorization = $7*11*13 = 1001$

Next, divide the given number to its prime factorization:

$1003003001/1001 = 1002001$

Repeating the process by dividing it to its proper prime factorization... $1002001/1001 = 1001$

then $1001/1001 = 1$

Therefore: $1003003001 = 7^3 \times 11^3 \times 13^3$ and $N = 3 + 3 + 3 = 9$

you have to prime factorise it. it is hard i know but. you can use a calculator if you want. But what is the divisor ? First it's 7 Like this carry on till you have got a less number. Then since you have three tries you can guess if you like it. But it's luck The correct answer is 9.

$1003003001$ is clearly not even, nor is it divisible by $3$ because its digit sum is not and it isn't divisible by $5$ . Now $7^{3} \times 2924207= 1003003001$ and $7$ doesn't divide $2924207$ , but continuing dividing by the next prime as many times as possible we find $1003003001 = 7^{3} \times 11^{3} \times 13^{3}$ which is nine primes.

1003003001=1000^3+3 1000^2 1^2+3 1000^1 1^2+1^3=(1000+1)^3 1001=13 11 7 ==> 1003003001=13^3 11^3 7^3