Question

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

Answer 1

Hint: Write the greatest 4-digit number using the greatest digit four times. Then find the prime factorization of the obtained greatest 4-digit number.

Complete step-by-step solution -
First, we have to find the greatest 4-digit number. We know that the greatest digit is ‘9’. So, we will form the greatest 4-digit number using ‘9’ four times.
i.e. the greatest 4-digit number will be 9999.
Now, we have to find out the prime factorization of 9999.
So, let us prime factorize 9999.
Steps used for prime factorization: -
We will start finding prime factors of 9999 by checking, starting from the smallest prime number. The smallest prime number is 2 but 9999 is not divisible by 2.
Then we will check the next smallest prime number which is 3. 9999 is divisible by 3 and $\dfrac{9999}{3}=3333$
So,
$\begin{align}
  & 3\left| \!{\underline {\,
  9999 \,}} \right. \\
 & \,\,\,\left| \!{\underline {\,
  3333 \,}} \right. \\
\end{align}$
Now, we have got 3333. It is also divisible by 3 and $\dfrac{3333}{3}=1111$.
$\begin{align}
  & 3\left| \!{\underline {\,
  9999 \,}} \right. \\
 & 3\,\left| \!{\underline {\,
  3333 \,}} \right. \,\, \\
 & \,\,\,\left| \!{\underline {\,
  1111 \,}} \right. \\
\end{align}$
Now, we have got 1111 and it is not divisible by 3. Next prime number is 5 and 1111 is not divisible by 5.
Next prime number is 7 and 1111 is not divisible by 7. Next prime number is 11 and 11 is divisible by 11 such that $\dfrac{1111}{11}=101$.
$\begin{align}
  & 3\left| \!{\underline {\,
  9999 \,}} \right. \\
 & 3\,\left| \!{\underline {\,
  3333 \,}} \right. \,\, \\
 & 11\left| \!{\underline {\,
  1111 \,}} \right. \\
 & \,\,\,\,\,\left| \!{\underline {\,
  101 \,}} \right. \\
\end{align}$
And 101 is itself a prime number. So, it won’t be divisible by any other prime number except itself.
So, we will get-
$\begin{align}
  & 3\left| \!{\underline {\,
  9999 \,}} \right. \\
 & 3\,\left| \!{\underline {\,
  3333 \,}} \right. \,\, \\
 & 11\left| \!{\underline {\,
  1111 \,}} \right. \\
 & \,101\,\left| \!{\underline {\,
  101 \,}} \right. \\
 & \,\,\,\,\,\,\,\,\,\,\,\,\,1 \\
\end{align}$
So, the prime factorization of $9999=3\times 3\times 11\times 101$.
Hence, the greatest 4-digit number is 9999 and its prime factorisation is $3\times 3\times 11\times 101$

Note: While writing a number as a product of its prime factor, make sure not to mention 1 in the factors. This is because 1 is neither a prime number nor a composite number.