Question

Real numbers
Class 10
Chapter - 1 -- Real numbers .

Explain
Euclid Division Lemma .
Explain in step by step

Give 2 example .
Explain your example .
​

Answer 1

AnswEr :

⠀⠀⠀⠀⠀❑ The Euclid division theorem tells us that if we have two positive integers a and b, then it is also possible to have unique whole numbers that prove a = bq + r, where 0 ≤ r < b.

⠀

$\frak{Here}\begin{cases}\sf{\:\;\; a = Dividend}\\\sf{\;\;\; b = Divisor}\\\sf{\;\;\; q = Quotient}\\\sf{\;\;\; r = Remainder}\end{cases}$

⠀⠀⠀

⠀⠀ Dividend = (divisor × quotient) + remainder

⠀

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

⠀

◗ Let's take two examples:

⠀

Example 1: Suppose we divide 117 by 14. Then, we get 8 as quotient and 5 as remainder.

⠀

$\frak{Here}\begin{cases}\sf{\:\;\; Dividend = 117}\\\sf{\;\;\; Divisor = 14}\\\sf{\;\;\; Quotient = 8}\\\sf{\;\;\; Remainder = 5}\end{cases}$

⠀

Clearly, 117 = (14 × 8) + 5.

⠀

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀⠀

Example 2: Suppose we divide 73 by 34. Then we get 2 as a quotient 5 as a reminder.

⠀

$\frak{Here}\begin{cases}\sf{\:\;\; Dividend = 73}\\\sf{\;\;\; Divisor = 34}\\\sf{\;\;\; Quotient = 2}\\\sf{\;\;\; Remainder = 5}\end{cases}$

⠀

Clearly, 73 = (34 × 2) + 5.

Answer 2

Answer:

${\underline{\boxed{\large{\rm{Euclid\: Division\: Lemma\: :-}}}}}$

$\mapsto$ Euclid Division Lemma defines that if two positive integers is "a" and "b", then there exists unique integers is "q" and "r", then it satisfies the condition and that will be,

$\leadsto$ a = bq + r = 0 [ where, 0 ≤ r < b ]

And, where,

• a = Dividend
• b = Divisor
• q = Quotient
• r = Remainder

Now, we understand this concept by taking two examples,

❶ A number when divided by 53 gives 34 as quotient and 21 as remainder. Find the number.

Solution :

Given :

• Divisor = 53
• Quotient = 34
• Remainder = 21

As we know that,

$\sf\boxed{\bold{\small{Dividend =\: (Divisor \times Quotient) + Remainder}}}$

Then, according to the question by using the formula we,

⇒ $\sf Dividend =\: (53 \times 34) + 21$

⇒ $\sf Dividend =\: 1802 + 21$

➠ $\sf\bold{\purple{Dividend =\: 1823}}$

$\therefore$ The required number is 1823.

_____________________________

❷ A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number.

Solution :-

Given :

• Divisor = 61
• Quotient = 27
• Remainder = 32

As we know that,

$\sf\boxed{\bold{\small{Dividend =\: (Divisor \times Quotient) + Remainder}}}$

Then, according to the question by using the formula we get,

↦ $\sf Dividend =\: (61 \times 27) + 32$

↦ $\sf Dividend =\: 1647 + 32$

➠ $\sf\bold{\purple{Dividend =\: 1679}}$

$\therefore$ The required number is 1679.