Question

If the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y, find y.​

Answer 1

Let us first find the HCF of 210 and 55.

Applying Euclid division lemna on 210 and 55, we get

210 = 55 × 3 + 45

55 = 45 × 1 + 10

45 = 4 × 10 + 5

10 = 5 × 2 + 0

We observe that the remainder at this stage is zero. So, the last divisor i.e., 5 is the HCF of 210 and 55.

∴ 5 = 210 × 5 + 55y

⇒ 55y = 5 - 1050 = -1045

∴ y = -19✔️✔️

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Answer 2

$\textbf{\underline{\underline{Answer :-}}}$

The value of y is -19.

$\textbf{\underline{\underline{Explanation :-}}}$

$\textsf{\underline{\underline{Given :}}}$

The two numbers = 210 and 5

HCF can be expressed as 210 x 5 + 55y

$\textsf{\underline{\underline{To Find :}}}$

The value of y

$\textsf{\underline{\underline{Solution :}}}$

First we need to find the HCF of 210 and 5.

Prime Factorization of 210 :

$\begin{array}{r|l} 2 & 210 \\\cline{1-2} 3 & 105 \\\cline{1-2} 5 & 35 \\ \cline{1-2} 7 & 7 \\\cline{1-2}& 1 \end{array}$

Prime Factorization of 5

$\begin{array}{r|l} 5 & 5 \\\cline{1-2}& 1 \end{array}$

$\implies$ 210 = 2 × 3 × 5 × 7

$\implies$ 5 = 5 × 1

HCF = 5

Solving the value of y :

$\boxed{\sf{210 \times 5 + 55y}}$

$\sf{\implies} \: 210 \times 5 + 55y = 5$

$\sf{\implies} \: 1050 + 55y = 5$

$\sf{\implies} \: 55y = 5 - 1050$

$\sf{\implies} \: 55y =- 1045$

$\sf{\implies} \: y =\dfrac{ - 1045}{55}$

$\sf{\implies} \: y =- 19$

$\boxed{\sf{y =- 19}}$

$\therefore$ The value of y is -19.

$\textbf{\underline{\underline{Verification :- }}}$

$\sf{\implies} \: 210 \times 5 + (55 \times ( - 19)) = 5$

$\sf{\implies} \: 210 \times 5+ ( - 1045) = 5$

$\sf{\implies} \: 1050 + ( - 1045) = 5$

$\sf{\implies} \: 5 = 5$

Both sides are equal.

$\therefore$ The value of y is -19.