The key here is to know that the remainder of a number by 10 is determined by it's unit digit. Even if you don't remember it, taking a few examples can also help establish it.
For example, 134 % 10 = 4 (which is the units digit in134 )
2566 % 10 = 6 (which is the units digit in2566 )
So we only need to determine the units digit of the term: (13^4n+3)(9^2n)
Let's rewrite it as (13^4n+3)(3^4n) (Since 9 = 3^2)
Let's find the pattern for 13 first:
n=1 => 13^n => 13 % 10 = 3
n=2 => 13^n => 169 % 10 = 9
n=3 => 13^n => __7 % 10 = 7
n=4 => 13^n => __1 % 10 = 1
n=5 => 13^n => __3 % 10 = 3
...
Notice that beyond 13^2, I don't need to calculate the value. I only used 169 because I remember that 13^2 is 169. I'm only interested in the units digit of 13^n so I just calculated what the units digit of 13^3 and and 13^4 would be.
Units digit of 13^3 = units digit of 169 => 9 x units digit of 13 => 9 x 3 => 27 => 7. Similarly, you can get unit digits
...
For example, 134 % 10 = 4 (which is the units digit in134 )
2566 % 10 = 6 (which is the units digit in2566 )
So we only need to determine the units digit of the term: (13^4n+3)(9^2n)
Let's rewrite it as (13^4n+3)(3^4n) (Since 9 = 3^2)
Let's find the pattern for 13 first:
n=1 => 13^n => 13 % 10 = 3
n=2 => 13^n => 169 % 10 = 9
n=3 => 13^n => __7 % 10 = 7
n=4 => 13^n => __1 % 10 = 1
n=5 => 13^n => __3 % 10 = 3
...
Notice that beyond 13^2, I don't need to calculate the value. I only used 169 because I remember that 13^2 is 169. I'm only interested in the units digit of 13^n so I just calculated what the units digit of 13^3 and and 13^4 would be.
Units digit of 13^3 = units digit of 169 => 9 x units digit of 13 => 9 x 3 => 27 => 7. Similarly, you can get unit digits
...
Statistics : Posted by siddhantvarma • on 30 Jun 2024, 23:57 • Replies 2 • Views 147
