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30 Jun 2024, 23:57
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
69% (01:43) correct
31%
(02:03)
wrong
based on 433
sessions
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Date
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For a positive integer n, what is the remainder when \((13^{4n+3})(9^{2n})\) is divided by 10?
A. 1
B. 3
C. 5
D. 7
E. 9
A. 1
B. 3
C. 5
D. 7
E. 9
01 Jul 2024, 00:02
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sahilsah456
CONCEPT: Remainder when N is divided by 10 is the same as the Unit digit of N
Now, unit digit of (13^4n+3)(9^2n) = (3^3)*(9^2n) = 7*1 = 7
CONCEPT: Cyclicity of Unit digit of 3 is 4 i.e. Unit digit repeats after 4th power hence Unit of 13^4n+3 = Unit of 3^4n+3 = Unit of 3^3 = 7
Cyclicity of Unit digit of 9 is 2 i.e. Unit digit repeats after 2nd power... so Unit digit of 9^2n = Unit digit of 9^2 = 1
Answer: Option D
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04 Jul 2024, 07:56
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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 in 134)
2566 % 10 = 6 (which is the units digit in 2566)
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 of number of the form 13^n.
Now units digit of 13^4n+3 = units digit of 13^3 => 7.
Similarly, we can find the cyclic nature of 3^4n which would tell us that units digit of 3^4n is 1.
Therefore, unit digit of (13^4n+3)(9^2n) is 7, which will be the remainder when you divide it by 10. Hence, D.
For example, 134 % 10 = 4 (which is the units digit in 134)
2566 % 10 = 6 (which is the units digit in 2566)
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 of number of the form 13^n.
Now units digit of 13^4n+3 = units digit of 13^3 => 7.
Similarly, we can find the cyclic nature of 3^4n which would tell us that units digit of 3^4n is 1.
Therefore, unit digit of (13^4n+3)(9^2n) is 7, which will be the remainder when you divide it by 10. Hence, D.
