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number theory class 11

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Chill Karo
Indian Certificate of Secondary Education (ICSE), New Delhi

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Chapter 1 What Is Number Theory? Number theory is the study of the set of positive whole numbers 1, 2, 3, 4, 5, 6, 7, . . . , which are often called the set of natural numbers. We will especially want to study the relationships between different sorts of numbers. Since ancient times, people have separated the natural numbers into a variety of different types. Here are some familiar and not-so-familiar examples: odd 1, 3, 5, 7, 9, 11, . . . even 2, 4, 6, 8, 10, . . . square 1, 4, 9, 16, 25, 36, . . . cube 1, 8, 27, 64, 125, . . . prime 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, . . . composite 4, 6, 8, 9, 10, 12, 14, 15, 16, . . . 1 (modulo 4) 1, 5, 9, 13, 17, 21, 25, . . . 3 (modulo 4) 3, 7, 11, 15, 19, 23, 27, . . . triangular 1, 3, 6, 10, 15, 21, . . . perfect 6, 28, 496, . . . Fibonacci 1, 1, 2, 3, 5, 8, 13, 21, . . . Many of these types of numbers are undoubtedly already known to you. Others, such as the modulo 4 numbers, may not be familiar. A number is said to be congruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4, and similarly for the 3 (modulo 4) numbers. A number is called triangular if that number of pebbles can be arranged in a triangle, with one pebble at the top, two pebbles in the next row, and so on. The Fibonacci numbers are created by starting with 1 and 1. Then, to get the next number in the list, just add the previous two. Finally, a number is perfect if the sum of all its divisors, other than itself, adds back up to the

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