Question

If the HCF of $ 210 $ and $ 55 $ is expressible in the form $ 210 \times 5 + 55y, $ find y.

Answer 1

Hint: Here we will use the Euclid’s Division Lemma theorem, where if we have two positive integers a and b, then there would be whole numbers q and r which satisfies the equation: $ a = bq + r, $ where $ 0 \leqslant r \leqslant b $ . Also “a” is the dividend and “b” is the divisor.

Complete step-by-step answer:
First of all let us find HCF (Highest common multiple) of $ 210 $ and $ 55 $
When applying the Euclid’s division Lemma on the given two numbers, we get –
 $ 210 = 55 \times 3 + 45 $
Since the remainder is $ 45 $ and is a non-zero number. So, we can now apply division lemma on the divisor and the remainder to get –
 $ 55 = 45 \times 1 + 10 $
Let us consider the divisor $ 45 $ and the remainder $ 10 $ and then apply division lemma to get –
  $ 45 = 4 \times 10 + 5 $
Similarly, let us consider the divisor $ 10 $ and the remainder $ 5 $ and then apply division lemma to get –
  $ 10 = 5 \times 2 + 0 $
Now, we can observe that the remainder at this stage is zero. So, that last divisor i.e. $ 5 $ is the HCF of the given numbers $ 210 $ and $ 55. $
Therefore, take the given expression-
 $ \therefore 5 = 210 \times 5 + 55y $
Make an unknown variable “y” the subject and others on one side of the equation.
 $ \Rightarrow 5 - \left( {210 \times 5} \right) = 55y $
Simplify the above equation –
 $ \Rightarrow 5 - \left( {1050} \right) = 55y $
When you subtract a bigger number from the smaller number, it gives the resultant negative number.
 $ \Rightarrow - 1045 = 55y $
The above equation can be re-written as –
 $ \Rightarrow 55y = - 1045 $
When the number in multiplicative on one side moves to opposite side then it goes to the division.
 $ \Rightarrow y = \dfrac{{ - 1045}}{{55}} $
Find the factors and then simplify the above fraction –
 $ \Rightarrow y = \dfrac{{ - 19 \times 55}}{{55}} $
Common factors from the numerator and the denominator cancel each other.
 $ \Rightarrow y = ( - 19) $ is the required solution.
So, the correct answer is “- 19”.

Note: Know the difference between the HCF and LCM properly and apply accordingly. HCF is the greatest or the largest common factor between two or more given numbers whereas the LCM is the least or the smallest number with which the given numbers are exactly divisible. Remember the concept of division and the multiples of the numbers at least till twenty.