FINDING THE REMAINDER IN DIVISIONS INVOLVING POWERS OF NUMBERS

FINDING THE REMAINDER IN DIVISIONS INVOLVING POWERS OF NUMBERS

Q. Find the remainder when 2^58 is divided by 7?

Ans. Let us find the pattern that remainders follow when successive powers of 2 are dividedby 7.

Remainder of 2^1 /7 is 2

Remainder of 2^2 /7 is 4

Remainder of 2^3 /7 is 1

Remainder of 2^4 /7 is 2

We can see that remainders are repeating after 3 steps, so the cyclicity is 3

Divide the power i.e. 58 by cyclicity which is 3. Remainder obtained is 1 when 58/3

So we can say that remainder when 2^58 is divided by 7 is same as the remainder of 2^1/7 which is 2.

• Some Basic Formulae:

1.      1. Identities

1. (a + b)(a - b) = (a2 - b2)
2. (a + b)2 = (a2 + b2 + 2ab)
3. (a - b)2 = (a2 + b2 - 2ab)
4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
5. (a3 + b3) = (a + b)(a2 - ab + b2)
6. (a3 - b3) = (a - b)(a2 + ab + b2)
7. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
8. When a + b + c = 0, then a3 + b3 + c3 = 3abc.

2.      The sum of first n natural numbers is = [n(n+1)]/2. 

Example: The sum first 49 natural number is = [49(49+1)]/2 = [49(50)]/2 = 49 * 25 = 1225

3.     The sum of the squares of the first n natural numbers is = [n(n+1)(2n+1)]/6.

Example: The sum of the squares of the first 10 natural numbers is = 10(11)(21)/6 = 385.

4.  The sum of the cubes of the first n natural numbers is = [n(n+1)/2]²

    Example: The sum of the cubes of the first 10 natural numbers is = [10(10+1)/2]²

= [(10*11)/2]² = 55² = 3025

5.       The sum of the first n odd natural numbers is = n²

Example: The sum of the first 24 odd natural numbers is = 24² = 576.

6.       The sum of the first n even natural numbers is = n(n+1)

Example: The sum of the first 10 even natural numbers is = 10(10 + 1) = 10 * 11 = 110

7.       The p/q form of any recurring number = The non-recurring and recurring part written once)-(the non-recurring part)/ As many 9’s as the number of digits in the recurring part followed by as many 0’s as digits in the non-recurring part.

Example: : Express 0.5858585….. as fraction

= (585- 5/(990)

=58/99