Question

find the least number which is exactly divisible by 16 24 and 36​

Answer 1

❙❙ ❈✪Answer✪❈ ❙❙

The least number which is divisible by 16, 24 and 36 is 144 .

▌✿⍟Explanation⍟✿▐

To do this sum we have to find the LCM of the 3 given numbers.

➢ LCM= Least Common Multiple.

To find the LCM first of all you have to find the factors of the 3 given numbers.

So,

Factor of :-

16 = (2×2×2×2)

24=2×2×2×3

36=2×2×(3×3)

So, the least common multiple (LCM) will be (2×2×2×2×3×3)

That's equal to 144

Note :-

● What is LCM?

➢LCM is the least common number or you may say multiple that can be used to divide all the given numbers.

✎Hope it helps

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Answer 2

$\Large\boxed{\bf{\quad144\quad}}$

Let the least number be x.

Since x is exactly divisible by 16,

• $\sf{x=16p}$

• $\sf{x=2^4\,p\quad\quad\dots(1)}$

Since x is exactly divisible by 24,

• $\sf{x=24q}$

• $\sf{x=2^3\times3q\quad\quad\dots(2)}$

Since x is exactly divisible by 36,

• $\sf{x=36r}$

• $\sf{x=2^2\times3^2\,r\quad\quad\dots(3)}$

Equating (1), (2) and (3),

$\longrightarrow\sf{2^4\,p=2^3\times3q=2^2\times3^2\,r}$

Dividing each by $\sf{2^2,}$ we get,

$\longrightarrow\sf{2^2\,p=2\times3q=3^2\,r}$

$\longrightarrow\sf{4p=6q=9r}$

We have to divide each by the LCM of 4, 6 and 9, i.e., 36.

$\begin{tabular}{r|c}&\sf{4,\ 6,\ 9}\\\cline{2-}\sf{2}&\sf{2,\ 3,\ 9}\\\cline{2-}\sf{3}&\sf{2,\ 1,\ 3}\\\cline{2-}\end{tabular}$

So,

$\longrightarrow\sf{\dfrac{4p}{36}=\dfrac{6q}{36}=\dfrac{9r}{36}}$

$\longrightarrow\sf{\dfrac{p}{9}=\dfrac{q}{6}=\dfrac{r}{4}}$

Therefore,

$\longrightarrow\sf{p:q:r=9:6:4}$

For least x, let,

• $\sf{p=9}$

• $\sf{q=6}$

• $\sf{r=4}$

Then x is,

$\longrightarrow\sf{x=16p=16\times9}=\bf{144}$

$\longrightarrow\sf{x=24q=24\times6}=\bf{144}$

$\longrightarrow\sf{x=36r=36\times4}=\bf{144}$