Exercise 51

Evaluate the following sums and products in modular arithmetic.

• (a)Â Â 21 +26 15, Â Â Â Â 21 Ã—26 15.

• (b)Â Â 19 +33 14, Â Â
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The Division Algorithm tells us that all the possible remainders on division by an integer n lie in the set

We denote this set by Author(s): The Open University

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

Course image: Stuart Rankin in Flickr made available under Creative Commons Attribution-NonCommercial 2.0 Licence.

All materials included in this course are
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Now try the quiz and see if there are any areas you need to work on.

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Question 1

Find the volumes of these objects.

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An important idea when calculating volumes of simple shapes is that of a cross-section. In the case of the rectangular box considered above, it is possible to slice through the box horizontally so that the sliced area is exactly the same as the area of the base or top; in other words, the areas of the horizontal cross-sections are equal.

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What is a volume? The word usually refers to the amount of three-dimensional space that an object occupies. It is commonly measured in cubic centimetres (cm3) or cubic metres (m3).

A closely related idea is capacity; this is used to specify the volume of liquid or gas that a container can actually hold. You might refer to the volume of a brick and the capacity of a jug â€“ but not vice versa. Note that a container with a particular volume will not nec
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Question 1

Find the area of a circle of (a) radius 8 cm, and (b) radius 15 m.

• (a)

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Question 1

Find the area of each of these shapes.

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You may like to add the area formulas in this section to your notes for future reference.

The simplest areas to find are those of rectangles. The area of a rectangle is its length multiplied by its breadth. Sometimes the dimensions of a rectangle are referred to as the base and the height, instead of the length and the breadth. The area is then expressed as the base multiplied by the height.

Question 1

Which of these triangles are similar?

Drawing circles freehand often produces very uncircle-like shapes! If you need a reasonable circle, you could draw round a circular object, but if you need to draw an accurate circle with a particular radius, you will need a pair of compasses and a ruler. Using the ruler, set the distance between the point of the compasses and the tip of the pencil at the desired radius; place the point on the paper at the position where you want the centre of the circle to be and carefully rotate the compass
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We have seen in SAQ 18 of Section 3.4 how some sets of points of the complex plane can be described algebraically in terms of operations on C. We now use the modulus function to take this a step further by defining discs in the complex plane. As we shall see, discs are extensively used in arguments involving l
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After studying this course, you should be able to:

• understand the definition of a real function

• use the notation for intervals of the real line

• recognise and use the graphs of the basic functions described in the audio section

• understand the effect on a graph of translations, scalings, rotations and reflections

• understand how the shape of a graph of a function features properties of the function such as increasing, decr
Author(s): The Open University

World in transition: Managing Resources
Do you take your access to water for granted? The Peruvian and Tanzanian communities featured in this album certainly donâ€™t. This album examines how development agencies can empower communities to help themselves by introducing simple technologies, and facilitate the sharing of ideas through education. In the Andean mountains, scarce supplies of water and agricultural challenges give rise to conflict; but the changes engineered by development agencies can start to show a way out of poverty. Me
Author(s): The OpenLearn team

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