Question

Find the H.C.F of 1405, 1465 and 1530 using Euclid division algorithm.

Answer 1

Hint: We should know how to find the H.C.F of three numbers a, b and c using the Euclid division algorithm. Now we should find the two largest numbers among a, b and c. Let us assume a is the largest number and c is the smallest number. We should obtain a relation between a and b with the equation \[a=bq+r\] where “q” is quotient and “r” is remainder. Now in the next step assume “q” obtained as “a” and “r” obtained as “b”. This process is continued until the value of r reaches zero. The value of q where the value of r is zero is termed as H.C.F of a and b. Let us assume this number as d. Now we should find the H.C.F of c and d using the above method. Let us assume this as e. This gives the H.C.F of a, b and c. By using this method, we should find the H.C.F of 1405, 1465 and 1530.

Complete step-by-step answer:
Before solving the question, we should know how to find the H.C.F of three numbers a, b and c using the Euclid division algorithm. Now we should find the two largest numbers among a, b and c. Let us assume a is the largest number and c is the smallest number. We should obtain a relation between a and b with the equation \[a=bq+r\] where “q” is quotient and “r” is remainder. Now in the next step assume “q” obtained as “a” and “r” obtained as “b”. This process is continued until the value of r reaches zero. The value of q where the value of r is zero is termed as H.C.F of a and b. Let us assume this number as d. Now we should find the H.C.F of c and d using the above method. Let us assume this as e. This gives the H.C.F of a, b and c.
Among three numbers, 1465 and 1530 are the two largest numbers among 1405, 1465 and 1530.
Now by using the Euclid division algorithm, we should find H.C.F of 1465 and 1530.
\[\begin{align}
  & 1530=1\times 1465+65 \\
 & 1465=65\times 22+35 \\
 & 65=35\times 1+30 \\
 & 35=30\times 1+5 \\
 & \text{30=5}\times \text{6+0}....\text{(1)} \\
\end{align}\]
From equation (1), we can say that the highest common factor of 1465 and 1530 is 5.
The smallest number among 1405, 1465 and 1530 is 1405.
Now we have to find the H.C.F of 5 and 1405.
\[1405=281\times 5....(2)\]
So, we can say that the H.C.F of 1405, 1465 and 1530 is 5.

Note: This problem can be solved in an alternative method also.
Now by using the Euclid division algorithm, we should find H.C.F of 1465 and 1530.
\[\begin{align}
  & 1530=1\times 1465+65 \\
 & 1465=65\times 22+35 \\
 & 65=35\times 1+30 \\
 & 35=30\times 1+5 \\
 & \text{30=5}\times \text{6+0}....\text{(1)} \\
\end{align}\]
From equation (1), we can say that the highest common factor of 1465 and 1530 is 5.
The smallest number among 1405, 1465 and 1530 is 1405.
We know that 1405 is greater than 5. So, we can say that 5 is the H.C.F of 1405, 1465 and 1530.