Question

A trader has 3 bundles of string 392m, 308m and 490m long. What is the greatest length of string that the bundles can be cut up into without any left over string?

Answer 1

Hint: The greatest length of string that the bundles can be cut up into without any left over will be the H.C.F or highest common factor of the three given lengths. To find the H.C.F, first of all use prime factorization to write the prime factors of 392, 308 and 490. Now, look for the factors which are common in all three. Multiply these common factors to get the answer.

Complete step by step answer:
We have been given three bundles of strings having length 329m, 308m and 490m and we have to find the greatest length of string that the bundles should be cut so that no string is left. So, this greatest length will be the H.C.F of the lengths of the given 3 strings.
Let us find the H.C.F of 392, 308 and 490. Breaking these numbers into the product of their prime factors, we get,
\[\begin{align}
  & 392=2\times 2\times 2\times 7\times 7 \\
 & 308=2\times 2\times 7\times 11 \\
 & 490=2\times 5\times 7\times 7 \\
\end{align}\]
Now, we know that H.C.F of a given set of numbers are the product of common factors present in all of them. So, the H.C.F of 392, 308 and 490 will be the product of all the common factors present in these three numbers.
Clearly, we can see that the factors that are common in all these three numbers are 2 and 7.
So, the required H.C.F = \[2\times 7\] = 14

Hence, the required greatest length of string that the bundles can be cut up into without any left over string is 14m.

Note: One must not get confused in L.C.M and H.C.F. Here, we do not have to find the L.C.M or least common multiple of the given numbers because if we will do so then we will get a number greater than all these 3 numbers which can never be the answer because we are cutting the strings and therefore, the length of each piece of string must be less than 308, 392 and 490.