number theory test

Question 1 - Question: Show that the sum of the cubes of any number of consecutive integers is divisible by the sum of those integers. Answer: Let the consecutive integers be (n + 1), (n + 2), (n + 3),....,(n + m) Sum of the cubes of integers = S 1 (n+m) and (n+m+1) are consecutive intege..

Question 1 - Question: Define an even number. Answer: A number which is divisible by 2 is called an even number..

Question 1 - Question: For the given sequence, suggest possible next three terms and find the general term. 7, 11, 15,---- Answer: First term = 7, Common Difference = 11 - 7 = 15 - 11 = 4 Next three terms are 15 + 4 = 19, 19 + 4 = 23, 23 + 4 = ..

A prime number is one that has only two factors namely 1 and itself. and a composite number has factors besides 1 and itsel..

The union of the set of rational numbers and irrational numbers forms the set of real numbers. (i) For every real number, there is a corresponding point on the number line. (ii) For every point on the number line, there exists a real..

Square root of a negative number is known as an imaginary number . If x and y are real numbers, then x + iy is called a complex number . x is called the real part and y is called the imaginary part . The following are the types of compl..

The union of the set of rational numbers and irrational numbers forms the set of real numbers. Q = {rational numbers} = {irrational numbers} Then = R = {real numbers..

It has factors other than one and itself, e.g., 21 is a composite number..

It can only be divided by one or itself i.e., a prime number has no factors other than 1 and itself. e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, are examples of prime numbers..

Rational Numbers - We have defined rational numbers as those which can be expressed as fractions: We now define decimal fractions. The rational numbers can be expressed as terminating or recurring decimals. Decimal fraction has denominator as 10 or a power of ..