Question

Find the largest number that divides 455 and 42 with the help of a division algorithm.

Answer 1

Hint:To solve this problem, first divide the largest number with smallest number and calculate the remainder. Then divide the divisor by the remainder and find the remainder again. Repeat this procedure till there is zero in remainder.

Complete Step-by-step solution
We know that according to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition \[a = bq + r\;{\text{ }}\] where \[0\; \leqslant r < b\;{\text{ }}\]
i) Start with the larger integer, 455

 ∴\[455 = \left( {42 \times 10} \right) + 35\;\;\] (By Euclid's lemma)

Now divide 42 by 35
ii) Consider 42÷35

∴\[42 = \left( {35 \times 1} \right) + 7\;\;{\text{ }}\]

     Now divide 35 by 7

Finally we got remainder as zero
Hence the HCF(455,42)=HCF(42,35)=HCF(35,7)=7
∴7 is the largest number that divides 455 and 42.

Additional information:
Additional information: The basis of the Euclidean division algorithm is Euclid’s division lemma. To calculate the Highest Common Factor (HCF) of two positive integers a and b we use Euclid’s division algorithm. HCF is the largest number which exactly divides two or more positive integers. That means, on dividing both the integers a and b the remainder is zero.

Note:Euclid's Division Algorithm states that the divided is equal to product of the divisor and quotient added to the remainder. Where, Quotient is denoted by q and the remainder is denoted by r. HCF * LCM = PRODUCT OF THE TWO NUMBERS