Question

Solve the following:
(a) Find the lowest number which leaves 4 as remainder when divided by 9 and 12.
(b) Find the lowest number which being increased by 3 is exactly divided by 9 and 12.

Answer 1

Hint: In this question, we first need to find the L.C.M of 9 and 12. Then adding 4 to that L.C.M gives the least number that leaves 4 as remainder. Now, again subtracting 3 from the L.C.M of 9 and 12 gives the least number that is divisible by 9 and 12 by increasing 3.

Complete step-by-step answer:
Now, from the given question we need to find the numbers that are exactly divisible by 9 and 12
L.C.M(Least Common Multiple)
The least number which is exactly divisible by two or more given numbers is called the L.C.M of those numbers.
Factorization method to find L.C.M
(1) Find the prime factors of each of the given number
 (2) Find the product of all prime factors which appears greatest number of time in prime factorization of any given numbers. The product is the required L.C.M
As we already know that the least number which will be divisible by both 9 and 12 will be the L.C.M of those two numbers.
Let us now write the factorization of the numbers 9 and 12
\[9=3\times 3\]
\[12=3\times 4\]
Now, the L.C.M of 9 and 12 can be written as
\[\Rightarrow L.C.M=3\times 3\times 4\]
Now, on further simplification we get,
\[\therefore L.C.M=36\]
Now, from the first part to get the least number that leaves remainder 4 when divided by 9 and 12 we need to add 4 to the L.C.M
\[\Rightarrow L.C.M+4\]
Now, on further substitution of the value we get,
\[\begin{align}
  & \Rightarrow 36+4 \\
 & \Rightarrow 40 \\
\end{align}\]
Thus, 40 is the least numbers that leaves remainder as 4 when divided by 9 and 12.
Now, from the second part to get the least number when increased by 3 when divided by 9 and 12 we need to subtract 3 from the L.C.M
\[\Rightarrow L.C.M-3\]
Now, on substituting the respective value we get,
\[\begin{align}
  & \Rightarrow 36-3 \\
 & \Rightarrow 33 \\
\end{align}\]
Thus, 33 is the least number which on increasing by 3 is exactly divisible by 9 and 12
Hence, 40 and 33 are the numbers.

Note: It is important to note that the least number that is divisible by 9 and 12 will be 36. Now, on adding 4 to it and then dividing it leaves that extra 4 as remainder and on subtracting 3 from 36 gives a number which will be divisible by 9 and 12 on increasing 3.
Instead of using the L.C.M to find the least number divisible by 9 and 12 we can also find it by checking the numbers which are multiples to both of them and choose the least one out of them. But it would be a bit lengthy.