Q.1. Determine the H.C.F of the following numbers by using Euclid's algorithm (ix):
(i) 300,450
(ii) 399,437
(iii) 1045,1520
Ans:
(i) 300 and 450
Dividend = 450 and divisor = 300
Clearly, the last divisor is 150.
Hence, HCF of the given numbers is 150.
(ii) 399 and 437
We have dividend = 399 and divisor = 437
Clearly, the last divisor is 19.
Hence, HCF of the given numbers is 19.
(iii) 1045 and 1520
We have dividend = 1045 and divisor = 1520
Clearly, the last divisor is 95.
Hence, HCF of the given numbers is 95.
Q.2. Show that the following pairs are coprime:
(i) 59,97
(ii) 875,1859
(iii) 288,1375
Ans: We know that two numbers are coprimes if their HCF is 1.
(i) 59 and 97
Here, dividend = 97 and divisor = 59
Clearly, the last divisor is 1.
Hence, the given numbers are coprimes.
(ii) 875 and 1859
Here, dividend = 1,859 and divisor = 875
Clearly, the last divisor is 1.
Hence, the given numbers are coprimes.
(iii) 288 and 1375
Here, dividend = 288 and divisor = 1,375
Clearly, the last divisor is 1.
Hence, the given numbers are coprimes.
Q.3. What is the H.C.F of two consecutive numbers?
Ans:
The HCF of two consecutive numbers is 1.
Example:
D = 4 and d = 5 are two consecutive numbers.
Here, we have dividend = 5 and divisor = 4
Clearly, the last divisor is 1.
Hence, HCF of 4 and 5 is 1.
Q.4. Write true (T) of false (F) for each of the following statements:
(i) The H.C.F of two distinct prime numbers is 1.
(ii) The H.C.F of two coprime number is 1.
(iii) The H.C.F of an even and an odd numbers is 1.
(iv) The H.C.F of two consecutive even numbers is 2.
(v) The H.C.F of two consecutive odd numbers is 2.
Ans:
(i) True.
(ii) True.
(iii) False.
HCF of 6 and 9 is 3, not 1.
(iv) True.
(v) False.
HCF of two consecutive odd number is 1.
Example: HCF of 25 and 27 is 1.
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