Problems on Euclids division algorithm

Question 31

Question:  

Answer:   

Question 32

Question:  

Answer:   

Question 33

Question:  

Answer:   

Let




The statement in the second pair of brackets is impossible. Thus, we arrive at a contradiction. The assumption is false and hence the theorem is true.

Question 34

Question:   Using Euclid's division algorithm, find the HCF of the following numbers. 624, 936, 264

Answer:    Using Euclid's division algorithm HCF (936, 624)
936= 624 x 1 + 312
624 = 312 x 2 + 0
HCF (936, 624) = 312
HCF (312, 264)
312 = 264 x 1 + 48
264 = 48 x 5 + 24
48 = 24 x 2+ 0
HCF (312, 264) = 24

Using long division
i) 624, 936, 264



HCF (936, 624) = HCF (624, 312) = 312

Now take 312 and 264


HCF of 624, 936 and 264 is 24.

Question 35

Question:   Using Euclid's division algorithm, find the HCF of the following numbers. 2103, 9945, 9216

Answer:   



Hence HCF of 9945, 9216 and 2103 is 3.

Question 36

Question:   Using Euclid's division algorithm, find the HCF of the following numbers. 13620, 221, 6810

Answer:   

HCF (6810, 221) = 1
HCF (13620, 6810, 221) = 1 which shows that 6810 and 221 are co-prime.

Question 37

Question:   Using Euclid's division algorithm, find the HCF of the following numbers. 594, 1848, 792.

Answer:    594, 1848, 792
HCF (1848, 792)
1848 = 792 x 2 + 264
792 = 264 x 3 + 0
HCF (1848, 792) = 264
HCF (594, 264)
594 = 264 x 2 + 66
264 = 66 x 4 + 0
HCF (594, 264) = 66
HCF (594, 1848, 792) = 66

Question 38

Question:   Find the HCF and LCM of 144 and 180 by the prime factor method. [2 Marks]

Answer:    144 2 x 2 x 2 x 2 x 3 x 3
180 = 2 x 2 x 3 x 3 x 5
HCF (144, 180) = 2 x 2 x 3 x 3 = 36
LCM (144, 180) = 2 x 2 x 2 x 2 x 3 x 3 x 5 = 720

Question 39

Question:   Write the factor tree for the following numbers and write them as the product of primes i) 81 ii) 32 iii) 45 iv) 120 v) 200 vi) 100 [each 2 Marks]

Answer:    i) 81 = 3 x 3 x 3 x 3 = 3⁴


ii) 32 =2 x 2 x 2 x 2 x 2 = 2⁵


iii) 45 = 3 x 3 x 5 = 3² x 5


iv) 120 = 2 x 2 x 2 x 3 x 5 = 2³ x 3 x 5


v) 200 = 2 x 2 x 2 x 5 x 5 = 2³ x 5²


vi) 100 =2 x 2 x5 x 5 =2² x 5²

Question 40

Question:   Given HCF (117, 221) =13. Find LCM (117, 221) [2 Marks]

Answer:    HCF (117, 221) = 13
We know that the product of HCF and LCM of 2 numbers is equal to the product of the two numbers.
HCF x LCM = Product of the 2 numbers.
13 x LCM =117 x 221
LCM = = 1989
Therefore, LCM (117, 221) = 1989