By Armand Borel (auth.)

This booklet is a book in Swiss Seminars, a subseries of development in arithmetic. it truly is an increased model of the notes from a seminar on intersection cohomology thought, which met on the collage of Bern, Switzerland, within the spring of 1983.

This quantity provides an advent to the piecewise linear and sheaf-theoretic types of that idea as built via M. Goresky and R. MacPherson in *Topology *19 (1980), and in *Inventiones Mathematicae* seventy two (1983). whereas a few familiarity with algebraic topology and sheaf idea is believed, the notes contain a self-contained account of additional fabric on constructibility, derived different types, Verdier duality, biduality, and on stratified areas, that's utilized in the second one paper yet now not present in ordinary texts.

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**Example text**

Since d is the GCD of a and b, d ja and d jb, so that there are integers h and k which satisfy Eq. 7). We will now show that, in fact, h and k are relatively prime. h; k/. If r D 1, then, by definition, h and k are relatively prime and we are done. rd/ kr . a; b/. Therefore r D 1, so h and k are relatively prime. Theorem 6. a; b/ D ab: Proof. a; b/ D d . u; v/ D 1. dv/ D ab: Theorem 6 provides a direct link between the GCD and LCM of two arbitrary positive integers. Thus given either the GCD or LCM, one is able to find the other by a simple division of integers.

In the Division Algorithm, we have assumed that the divisor b is positive but here we made no assumption about the dividend a. What then is the interpretation of the Division Algorithm if a < 0? Let’s start with an example. Example 12. What is the quotient and remainder of 5 on division by 2? Solution. 10) where 0 Ä r < 2. In the interpretation of the case in which a and b are both positive, we used subtraction of b from a. However, subtraction of 2 from 5 gives 7. We are going in the wrong direction to get the quotient and remainder of 5 on division by 2.

Thus the remainder is 1 and the quotient is 3. Note that there is no assumption that the quotient must be positive in the Division Algorithm. 3: 1. (a) What are the quotient and remainder of 63 divided by 17? (b) Follow the discussion at the beginning of this section and first subtract 17 from 63, then subtract 17 from your answer, etc. until you end up with an integer which is at least 0 but less than 17. Show that the number of subtractions is the quotient of 63 by 17 and the remainder is the number between 0 and 17.