Question

To find the hcf of two numbers experimentally based on euclid division lemma practical notes

Answer 1

Activity to find the h.c.f. of two numbers experimentally based on euclid's division lemma or algorithm-----

Euclid’s division lemma: IT states that for any given two positive integers ‘a’ and ‘b’ we can find two whole numbers ‘q’ and ‘r’

 such that a = b × q + r  where 0 ≤ r < b.

It is used to find the highest common factor of any given two positive integers and also to depict the common properties of numbers.

The following steps to obtain H.C.F using Euclid’s division lemma:

1. Consider two positive integers ‘a’ and ‘b’ such that a > b.

2. Apply Euclid’s division lemma to the given integers ‘a’ and ‘b’ to find two whole numbers ‘q’ and ‘r’ such that, a = b x q + r.

3. Check the value of ‘r’. If r = 0 then ‘b’ is the HCF of the given numbers.

4. If r ≠ 0, apply Euclid’s division lemma to find the new divisor ‘b’ and remainder ‘r’.

5. Continue this process till the remainder  becomes zero. In that case the value of the divisor ‘b’ is the HCF (a , b). Also HCF(a ,b) = HCF(b, r). 

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Answer 2

Answer:

To find the HCF of two numbers experimentally based on Euclid's division lemma.

Material Required

Square sheet and maths kit.

Procedure

1. Cut one strip of length “a” units (a=11 cm) and one strip of length “b” units (b=8 cm) of width 1 cm each (a>b).

2. Paste the strips of length “a” units above the strip of length “b” units aligning them from length as shown in fig 1. The remaining length is say c cm (c=3 cm, b>c).

3. Cut another strip of length b units and 2 strips of c units. Paste the strip of b units above the strips of c unit aligning them from left ad shown in fig 2. The remaining length is say d cm. (d=2 cm, d<c)

4. Repeat the process till the length proceeding strips covered completely and second strip which covers the proceeding strips is the HCF of given number (fig 4).

Observation

By Euclid's division lemma a=bq+r, 0≤r<b

Fig 1. Shows a=b×1+c (q=1,r=c)

 

 

Fig 2. shows b=c×2+d (q=2,r=d)

Fig 3. shows c=d×1+e (q=1,r=e)

Fig 4. shows d=e×2+0 (q=2,r=0)

H.C.F. of a and b is e.

Here, a=11 cm, b=8 cm, c=3 cm, d=2 cm and e=1 cm

H.C.F of 11 and 8 is 1.

Conclusion

Euclid's division lemma can be used for finding the HCF of two or more numbers.

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