Question

Write the greatest 4 digit number and express it in the form of its prime factors.

Answer 1

Hint: The prime numbers which are multiplied together to give the original number are said to be the prime factors of the given number. Just recall some tests of divisibility before proceeding the problem.

Complete step-by-step solution -
 (1) Divisibility by 2: All the numbers whose unit digits is any of 0, 2, 4, 6, 8.
 (2)Divisibility by 3: If the sum of its digits is divisible by 3 then, the given number is also
  divisible by 3.
 (3)Divisibility by 5: The numbers whose unit digit is either 0 or 5, then it is divisible by 5.
 (4)Divisibility by 7: If the unit digit of a given number is doubled and subtracted from the
    rest, which gives the number divisible by 7, then the given number is also divisible by 7.
 (5)Divisibility by 11: A number is divisible by 11, if the difference of sum of the digits in its
   the odd place and sum of the digits in its even place is either 0 or number exactly divisible by 11.

Let us start our solution by writing the greatest 4 digit number i.e. 9999.
And, we need to express it in the form of its prime factors. Now, we shall find the least prime number which divides 9999 exactly.
Let us write some prime numbers in increasing order, so that we can check whether those prime numbers divide 9999 exactly.
 2, 3, 5, 7, 11, 13, 17….
Now, we need to check whether 2 divides 9999 exactly. Since, 9999 is an odd number it is not divisible by 2. Then, we shall check with divisibility by 3. Since, addition of each digit gives 36 which is divisible by 3, 9999 is also divisible by 3. Now, we will express 9999 as follows.
$9999=3\times 3333.....(1)$
Then, do the same for 3333. Since, 3333 is an odd number we go for checking the divisibility by 3. On adding the digits of the number, we get 12, which is exactly divisible by 3. So, 3 divides 3333 exactly. Now, on expressing (1) we get,
$9999=3\times 3\times 1111.....(2)$
We know that 1111 is odd and the sum of the digits gives 4. So, it is not divisible by 2 and 3. Then, check the divisibility by 5 and 7. We can see that, unit digit of 1111 is neither 0 nor 5. So, 1111 is not exactly divisible by 5. Then, on considering the unit digit, doubling it and then subtracting it from the remaining, we get 109, which is not exactly divisible by 7. So, 1111 is not exactly divisible by 7. Then, check divisibility of 1111 by 11. We can see that, difference between the sum of its digits at odd places and the sum of its digits at even places is 0. So, 1111 is exactly divisible by 11. Now, on expressing (2) we get,
$9999=3\times 3\times 11\times 101.....(3)$
$11>\sqrt{101}$.
We know that prime numbers below 11 i.e. 2,3,5,7 neither of them divides 101 exactly. So, 101 is prime.
Since, 101 is a prime number no more factors can be given. So, we shall give the prime factors of 9999 as,
$9999=3\times 3\times 11\times 101$

Note: Always test the divisibility of a given number by the prime numbers one by one starting from the least. And, terminate when all of your factors are prime numbers. And, to check whether a given number, x is prime number, consider some y such that, $y>\sqrt{x}$ . Take all the prime numbers less than y and if none of them divides x exactly, then x is said to be a prime number.