I haven't taken calculus so please explain this without using (mod) 1.?

Question

What is the remainder when 3^(4n+ 2) + m is divided by 10

This is a data sufficiency question and one of the prompts is:

m = 1. This is sufficient. Please explain why. Without using (mod) 10. I have no idea what (mod) means

Answer 1

Modular arithmetic is not a part of calculus!
mod stands for modulo and we say that this is modular arithmetic.

All it is is that whatever you divide by ten, you are concerned with the remainder. So Z/Z10 (read Z mod Z 10) where z is the set of all integers has only 10 elements in it. Z mod Z 10 is a new number system with only 10 elements in it {0,1,2,3,4,5,6,7,8,9} because those are the only remainders possible when you divide an integer by 10.

So we say that
0=0 mod 10
1=1 mod 10
2=2 mod 10
3=3 mod 10
4=4 mod 10
5=5 mod 10
6=6 mod 10
7=7 mod 10
8=8 mod 10
9=9 mod 10
10=0 mod 10 (because 10/10 has a remainder of zero)
11=1 mod 10
12=2 mod 10
.
.
.
999=9 mod 10
1000=0 mod 10
10011111127641982736542 = 2 mod 10
and so on...

You can do this with any natural number. The reason mod 10 is so special because a number moded out by 10 gives us its units digit.

As for your question, it is simple.
3^(4n+2) + m expand this out.
3^(4n)*3^2 + m
(81^n)*9 + m

81^n will always end up with 1 in its units place which means that
(81^n)*9 will always end up with 9 in its units place.

You add m and you get the units place is the units place of 9+m.

And when you divide a number by 10, you gets its units place. So when you divide 3^(4n+2)+m by 10, the remainder will the remainder when 9+m is divided by 10. Since m=1, 9+m=9+1=10 which means that if m=1, the remainder is zero.

For any general m, the remainder will be the units place of 9+m.

So if I let m=23, 9+m=32 so the remainder will be 2.

Answer 2

To find the remainder of any integer when divided by 10, knowing the last digit of the integer will be data sufficient.

To find the remainder of the following when divided by 10:
m is insufficient
n + m is insufficient
0 x (n + m) is sufficient because without knowing the values of n and m, we know the result is 0.

0 x n + m is insufficient unless we know the value of m.

3^(4n + 2) + m is insufficient unless we know the value of m. E.g. if m = 1, then the remainder is also the remainder of 9+m = 9+1 = 10 when divided by 10, which is 0.

Why 9? (the following is sort of repeating the above answers)
3^(4n+2)
= 3^(4n) x 3^2
= (3^4)^n x 9
= 81^n x 9
Without knowing the value of n, we know that 81^n is a number ending with 1 because any integer z multiplies by another number with ending with a 1 will result in a value the end with the same digit as the integer z.
Such an integer when multiplied by 9 will end with the digit 9.

To find the remainder when divided by 10 of the sum of a number that ends with 9 and m is data insufficient unless m is known. So, when m is known, we would know the last digit of the value of 3^(4n + 2) + m, and thus it is data sufficient to find the remainder when divided by 10.

Answer 3

The modulus function (%) gives the remainder when two numbers are divided. From what you said about prompting, I am assuming you are talking about computer programming. I learned about modulus function in my C++ class. Here's an example of modulus:

26 % 3

This function, in C++, would divide 26 by 3, and display ONLY the remainder.

Now i'll try to simplify your expression for case when m=1:

[3^(4n+ 2) + 1] / 10

= [3^(4n) * 3^2 + 1] / 10

= [((3^4)^n) * 3^2 + 1] / 10

= [81^n * 9 + 1] / 10


I don't know if this can be simplified more. It probably can, but I suck at algebra.

Answer 4

The last digit of 3^4n is 1 for any integer n. So, the last digit of x^(4n+2) has to be 9 (i.e. 1*3*3).

Assuming m is an integer between 1 and 10, then the remainder is (m-1).

Answer 5

It is better that you take Pre-Cal first. You may not be lost in Calculus but you sure will be lost in Calculus 2 or 3. I know this because I am taking Cal 2 and you really need to know stuff from pre-cal. This is my opinion, Unless you like to study a lot haha then I don't think you would really need it but it would be better.

Answer 6

I knoooww my calculus!

It says u + me = us!

Yaaaa!!

I knoooww my calculus!