Question

Determine the largest $3$-digit number which is exactly divisible by $8,10$ and $12$.

Answer 1

STEP 1 : Finding the L.C.M. of $8,10$ and $12$ by division method

Using the division method we find the LCM of $8,10$ and $12$.

L.C.M. of $8,10$ and $12$ $=2\times 2\times 2\times 3\times 5=2^3\times 3\times 5$

L.C.M. of $8,10$ and $12$ $=8\times 3\times 5=120$

STEP 2 : Finding the largest $3$-digit number which is multiple of $120$

We know that largest $3$-digit number is $999$

Dividing $999$ by $120$ we get

$\Rightarrow 999=120\times 8+39$

If we subtract the remainder from $999$ we get the largest $3$-digit number which is multiple of $120$

$\Rightarrow 120\times 8=999-39=960$

i.e. $960$ is the largest $3$-digit number which is multiple of $120$.

Hence, the largest $3$-digit number which is exactly divisible by $8,10$ and $12$ is $960$.