Euclid's Division Lemma

Let $p$ and $q$ be two natural numbers such that $p>q$. When $p$ is divided by  $q$, the remainder is $r$.

$i)$ $r$ cannot be $\left(p-q\right)$

$ii)$ $r$ can either be $q$ or $\left(p-q\right)$

$iii)$ $r$ is definitely less than $q$

Which of the above statements is/are true?

The HCF of $k$ and $93$ is $31$, where $k$ is a natural number. Which of these can be true for some values of $k$?

$\left(\mathrm{i}\right)$ $k$ is a multiple of $31$

$\left(\mathrm{ii}\right)$$k$ is a multiple of $93$

$\left(\mathrm{iii}\right)$ $k$ is an even number.

$\left(\mathrm{iv}\right)$ $k$ is an odd number.

If a natural number $\text{'}a\text{'}$ is divided by $7$, the remainder is $5$. If a natural number $\text{'}b\text{'}$ is divided by $7$, the remainder is $3$. The remainder is $\text{'}r\text{'}$ if $a+b$ is divided by $7$. Find the value of $\frac{3r+5}{4}$.

Find the HCF of the following

$3,12$

_____

Find the L.C.M of $8$ and $24$

Atul exercises every $16$ days and Arjun exercises every $10$ days. Atul and Arjun both exercised today. How many days will it be until they exercise together again.

If the H.C.F of $\left(96,404\right)=4$, then find the L.C.M of $\left(96,404\right)$

Find the L.C.M of $336$ and $54$ using prime factorisation method

Find the L.C.M of $6,72, 120$ using prime factorisation method.

Identify the natural numbers among the given series of numbers: $\left\{-3,-10,0,5,8,4.6,98,2.9\right\}$

Are the natural numbers positive integers? Justify the answer.

Natural numbers are whole numbers. Justify the answer.

Negative numbers are natural numbers.

$0$ is a natural number

Find the HCF of $250$ and $75$ using Euclid's Division Lemma

Find HCF of $135$ and $225$ (use Euclid's division Algorithm).

$17=6\times 2+5$ is compared with Euclid's Division lemma $a=bq+r$, then which number is representing the remainder?

Find the HCF by using long division method:
$30,90,120$

Find the HCF by using long division method:
$175,105,70$.

Find the HCF by using long division method:
$240,510$