Question

Write the largest 4-digit number and give its prime factorisation.

Answer 1

Hint: Here, we will write the largest 4-digit number that is, 9999. Then, we will use the concept of prime factorisation of numbers. Prime factorisation is the method of finding the prime factors of the given number.

Complete step by step answer:
The largest number of 4 digits is the number 9999. The number that comes after 9999 is 10000, which is a 5-digit number.
Now, we will give the prime factorisation of 9999.
We know that in the number system, prime numbers are the numbers which have only two factors, 1 and the number itself. Hence, it is obvious that they will be divisible by 1 and the number itself.
Now, the number is 9999. Let us find its divisibility by the prime divisors, starting with the smallest prime number, which is 2.
We observe that the number 9999 is an odd number. Hence, it is not divisible by 2.
Next, we sum up the digits of the number 9999.
Hence, we get $$9+9+9+9=36$$.
We see that the sum of the digits of the given number 9999 is divisible by 3.
So, the number 9999 is divisible by 3.
Thus, dividing 9999 by 3, we get
$${\displaystyle \frac{9999}{3}}=3333$$
Next, we sum up the digits of the result 3333.
Hence, we get $$3+3+3+3=12$$.
We see that the sum of the digits of the given number 3333 is divisible by 3.
So, the number 3333 is divisible by 3.
Thus, dividing 3333 by 3, we get
$${\displaystyle \frac{3333}{3}}=1111$$
Now, the last digit of the number 1111 is neither 0 nor 5.
Thus, the number 1111 is not divisible by 5.
Now, we check the divisibility of the number by 7. To check divisibility by 7, twice the unit’s place digit subtracted from the rest of the number should be divisible by 7.
Here, twice the unit’s place digit is $$1\times 2=2$$.
We subtract 2 from the remaining number 111 to check whether 1111 is divisible by 7.
$$111-2=109$$
Thus, the number 1111 is not divisible by 7.
Now, we check the divisibility of the number by 11. To check divisibility by 11, the sum of the digits at even places is subtracted from the sum of the digits at odd places. The number is divisible by 11 if the result is divisible by 0 or 11.
Here, the sum of the digits at even places is $$1+1=2$$ and the sum of the digits at odd places is $$1+1=2$$.
Subtracting 2 from 2, we get the result $$2-2=0$$.
Hence, the number 1111 is divisible by 11.
Thus, dividing 1111 by 11, we get
$${\displaystyle \frac{1111}{11}}=101$$
The number 101 is a prime number.
Hence, resolving into factors, we get that 9999 can be expressed as
$$9999=3\times 3\times 11\times 101$$
In this expression, 9999 is expressed as a product of its prime factors 3, 3, 11 and 101.


Note:
A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits.For checking divisibility by smaller prime numbers, in most cases, it is not required to actually divide the number by the prime number, but we can apply the rules of divisibility effectively.