Question

Prove that for a positive integer n we have that 3^3n+3 -26n -27 is divisible by 169.

Answer 1

Step-by-step explanation:

Since n is a square, as well as divisible by 14 (NOTE: any number divisible by 14 is also divisible by 7), a very quick number to look at is 14 X 14 = 196.

Again, since n is a square, one needs to look at other squares lesser than 196. These will be squares of 13, 12, 11, 10, 9, and so on. Most of these can be immediately eliminated because they will not be divisible by 14.

Another way to approach:

1. N is a square: so it will be of the form m X m

2. N is a multiple of 14 (ie N is divisible by 14), which means 'm' must also be divisible by 14.

3. To find the "lowest" N, we need to find the lowest "m". The lowest value for 'm' which is divisible by 14, is, well, 14.

4. So, if 'm' is 14, then N = m X m = 14 X 14 = 196

A third way:

Simply find the square of 14; and then look at the choices you have (if it is a multiple-choice question). If there are values lower than 196, see if those are divisible by 14. Chances are only 1 other value may be; and that may not be a square.

873 views

View upvotes

5

Related Questions (More Answers Below)