Question

Find the smallest number which when increased by 17 is exactly divisible by both 520 and 468.

Answer 1

Hint: We start solving the problem by assuming the variable for the required smallest number. We then recall the property that the smallest number which can be divided by given two or more numbers is equal to its L.C.M (Least Common Multiple). We then find the L.C.M of the given two numbers and equate it to the result we obtain after adding 17 to the assumed variable. We then make the necessary calculations to get the required result.

Complete step-by-step answer:
According to the problem, we need to find the smallest number which is exactly divisible by both 520 and 468 when it is increased by 17.
Let us assume the smallest number is ‘x’.
We know that the smallest number which can be divided by given two or more numbers is equal to its L.C.M (Least Common Multiple).
So, the L.C.M of the numbers 520 and 468 will be the number when ‘x’ is increased by 17. This means that the L.C.M of the numbers 520 and 468 is $x+17$.
Let us first find the L.C.M of numbers 520 and 468 which is as follows.
$\begin{align}
  & 13\left| \!{\underline {\,
  520,468 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  40,36 \,}} \right. \\
 & \text{ }2\left| \!{\underline {\,
  20,18 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  10,9 \,}} \right. \\
\end{align}$.
Now, the L.C.M of the numbers 520 and 468 is $13\times 2\times 2\times 10\times 9=4680$.
So, we get $x+17=4680$.
$\Rightarrow x=4680-17$.
$\Rightarrow x=4663$.
So, we have got the value of ‘x’ as 4663.
∴ The smallest number which when increased by 17 is exactly divisible by both 520 and 468 is 4663.

Note: Whenever we get problems involving common numbers which are divisible by two or more numbers, we should try to make use of L.C.M. We can also calculate L.C.M by dividing the multiplication of given numbers with their G.C.D (Greatest Common Divisor). We should not make mistakes while finding the L.C.M of the given numbers. Similarly, we can expect the problems to find the number that is present between 10000 and 15000 which is exactly divisible by 520 and 468.