A proven statement used for proving another statement is called a LEMMA.
A series of well defined steps which gives a procedure for solving a type of problem is known as an ALGORITHM.
The word algorithm comes from the name of the 9th centuary Persian mathematician al-Khwarizmi. The word algebra is derived from a book, he wrote, called Hisab-al-jabr w'al-muqabala.
Euclid's division lemma: Given positive integers a and b, there exist unique integers q and r satisfying
\( a = bq + r \), 0 ≤ r ≤ b
Euclid's Division Algorithm is a technique to compute the HCF (Highest Common Factor) of two given integers.
Ealier computers are programmed as per Euclid's Division of Algorithm to compute HCF.
Euclid's Division Algorithm is stated for only positive integers, it can also be extended for all integers except zero ( b ≠ 0 )
Exercise 1.1
(Watch related video given below)
Question 1. Use Euclid's division algorithm to find the HCF of (i) 135 and 225, (ii) 196 and 38220, (iii) 867 and 255
Question 2. Show that any positive odd integer is of the form 6q + 1 , 6q + 3 or 6q + 5, where q is some integer.
Question 3. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Question 4. Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
Question 5. Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m , 9m + 1 , 9m + 8.