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

The largest 4-digit number is 9999.

To find the prime factorization of 9999, we start by dividing it by the smallest prime numbers.

9999 is an odd number, so it is not divisible by 2.

We check for divisibility by 3. The sum of the digits of 9999 is 9 + 9 + 9 + 9 = 36, which is divisible by 3. Therefore, 9999 is divisible by 3.

9999 ÷ 3 = 3333

Next, we check 3333 for divisibility by 3 again. The sum of the digits of 3333 is 3 + 3 + 3 + 3 = 12, which is divisible by 3. So, we divide 3333 by 3.

3333 ÷ 3 = 1111

Now, we check 1111 for divisibility by 3. The sum of the digits is 1 + 1 + 1 + 1 = 4, which is not divisible by 3. We check for divisibility by 11 next. The alternating sum of the digits is 1 - 1 + 1 - 1 = 0, which is divisible by 11. So, we divide 1111 by 11.

1111 ÷ 11 = 101

Finally, 101 is a prime number, so the factorization process stops here.

Thus, the prime factorization of 9999 is:

3 × 3 × 11 × 101

Or, more concisely:

3² × 11 × 101