Question

When an iron rod is cut into equal pieces of $30$ cm each, a piece of $4$ cm is left out. When cut into equal pieces of $29$ cm, a piece of $13$ cm is left out. The minimum length of rod is

• A. $270$ cm
• B. $272$ cm
• C. $274$ cm
• D. $280$ cm

Answer 1

The correct option is C $274$ cm

Let the length of rod be x.

If rod cuts in 30 cm each, 4cm from left

$⟹x=\left(n_1\right)\left(30\right)+4-------\left(1\right)$

If cuts in 29 cm each 13 cm left

$x=\left(n_2\right)\left(29\right)+13------------\left(2\right)$

$⟹n_1\left(30\right)+4=\left(n_2\right)\left(29\right)+13$

$\left(n_1\right)\left(30\right)=n_2\left(29\right)+9---------\left(3\right)$

From (1) , for different values of n,

$x=34,64,94,124,154,184,214,244,274,304,.......$

Similarly from (2)

$x=42,71,100,129,158,187,216,245,274,303,.......$

From above it is clear that $274$ is minimum common value.So, minimum length must be $274$ cm.