Problem Solving

I can't seem to grasp this answer..even with the answer...can someone help? thanks!

If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

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ANSWER:


To find the remainder when a number is divided by 5, all we need to know is the units digit, since every number that ends in a zero or a five is divisible by 5.

For example, 23457 has a remainder of 2 when divided by 5 since 23455 would be a multiple of 5, and 23457 = 23455 + 2.

Since we know that x is an integer, we can determine the units digit of the number 7^(12x+3) + 3. The first thing to realize is that this expression is based on a power of 7. The units digit of any integer exponent of seven can be predicted since the units digit of base 7 values follows a patterned sequence:

Units Digit = 7


Units Digit = 9


Units Digit = 3


Units Digit = 1
Since we see that the pattern repeats itself every 4 integer exponents.

The question is asking us about the 12x+3 power of 7. We can use our understanding of multiples of four (since the pattern repeats every four) to analyze the 12x+3 power.

12x is a multiple of 4 since x is an integer, so 7 12x would end in a 1, just like 74 or 78.
7 12x+3 would then correspond to 73 or 77 (multiple of 4 plus 3), and would therefore end in a 3.

However, the question asks about 712x+3 + 3.
If 7 12x+3 ends in a three, 7 12x+3 + 3 would end in a 3 + 3 = 6.

If a number ends in a 6, there is a remainder of 1 when that number is divided by 5.

The correct answer is B.

Are you sure that the question isn't:

What's the remainder when 7^(12x+3) + 3 is divided by 5?

If that's the question, then the explanation makes sense (since it talks about powers of 7).

oops.
yes it's 7^(12x+3) + 3

however, i'm not sure how it is decided that the units would be 3 rather than, 7,9, or 1.

This can be answered pretty fast this way....

7^(12x+3) + 3= [49^6x][7^3] + 3

Now 49 when divided by 5 instead of taking the remainder as 4 we can take it as -1 so that the calculation is made fast. Since -1 is raised to the power 6x, which is always even, we can conclude that the remainder when 49^6x is divided by 5 is 1. Also 7^3 when divided by 5, the remainder is 3.

So the remainder when 7^(12x+3) is divided by 5 is 3.

Hence the remainder when 7^(12x+3) + 3 is divided by 5 is 1.

ieeyorei wrote:oops.
yes it's 7^(12x+3) + 3

however, i'm not sure how it is decided that the units would be 3 rather than, 7,9, or 1.

I think the explanation given is probably easier to understand than suresh's (which is correct, but a lot more complicated than we need), so let's reexamine that one.

For powers of 7, the units digit goes through a 4 part cycle:

7

9

3

1

So, to answer the question, we need to figure out where in the cycle we are.

Using our multiplication of exponent rule, we can change

7^(12x+3)

to

(7^12x) * (7^3)

Now let's look at each piece.

7^3 is easy.. that's the 3rd part of the cycle, so units digit will be "3".

7^12x seems more complicated, but it really isn't. We can rewrite 12x as 4*3*x, which is definitely going to be a multiple of 4. Therefore, no matter the value of x, 12x will always hit the 4 spot in the cycle. Accordingly, 7^12x will have a units digit of "1".

So, we're multiplying a units digit of 3 by a units digit of 1, giving us a units digit of 3.

One more step: now we're adding 3 to the total, giving us a units digit of 6.

When we divide a number with a units digit of 6 by 5, we'll end up with a remainder of 1... choose (B).

Stuart Kovinsky wrote:

ieeyorei wrote:oops.
yes it's 7^(12x+3) + 3

however, i'm not sure how it is decided that the units would be 3 rather than, 7,9, or 1.

I think the explanation given is probably easier to understand than suresh's (which is correct, but a lot more complicated than we need), so let's reexamine that one.

For powers of 7, the units digit goes through a 4 part cycle:

7

9

3

1

So, to answer the question, we need to figure out where in the cycle we are.

Using our multiplication of exponent rule, we can change

7^(12x+3)

to

(7^12x) * (7^3)

Now let's look at each piece.

7^3 is easy.. that's the 3rd part of the cycle, so units digit will be "3".

7^12x seems more complicated, but it really isn't. We can rewrite 12x as 4*3*x, which is definitely going to be a multiple of 4. Therefore, no matter the value of x, 12x will always hit the 4 spot in the cycle. Accordingly, 7^12x will have a units digit of "1".

So, we're multiplying a units digit of 3 by a units digit of 1, giving us a units digit of 3.

One more step: now we're adding 3 to the total, giving us a units digit of 6.

When we divide a number with a units digit of 6 by 5, we'll end up with a remainder of 1... choose (B).

We can conclude quickly that 7^(12x+3) is 3 in the following way.

7^(12x+3) = (49)^6x * 7^3

Since 9^(even) always ends with 1, 49^6x ends with 1.

So 7^(12x+3) ends with 3.

sureshbala wrote: We can conclude quickly that 7^(12x+3) is 3 in the following way.

7^(12x+3) = (49)^6x * 7^3

Since 9^(even) always ends with 1, 49^6x ends with 1.

So 7^(12x+3) ends with 3.

I saw the first time you posted that! I even referenced it in my post!

YOU may be able to conclude the answer quickly based on that method, but a huge number of test takers don't make giant intuitive algebraic leaps, which is why I offered the alternative solution.

(As an aside, I'm not even sure why you'd choose to go that route, with your level of understanding of math it would be much quicker not to convert to powers of 49.)

Stuart Kovinsky wrote:

sureshbala wrote: We can conclude quickly that 7^(12x+3) is 3 in the following way.

7^(12x+3) = (49)^6x * 7^3

Since 9^(even) always ends with 1, 49^6x ends with 1.

So 7^(12x+3) ends with 3.

I saw the first time you posted that! I even referenced it in my post!

YOU may be able to conclude the answer quickly based on that method, but a huge number of test takers don't make giant intuitive algebraic leaps, which is why I offered the alternative solution.

(As an aside, I'm not even sure why you'd choose to go that route, with your level of understanding of math it would be much quicker not to convert to powers of 49.)

Dear Stuart,

I agree with you. Once the student understands the concept of this cyclicity he can conclude the answer fast. I was just trying to put another way of answering this question.

Anyway, as you said ...

For 2, 3, 7 and 8 the cyclicity is 4
For 4 and 9 it is 2
And coming to 1, 5 and 6 , irrespective of the non-negative powers to which they are raised they always end with 1, 5 and 6 respectively.

I agree with bala. I always employ this approach......these kind of questions are always constructed so that a remainder of 1 can be reached pretty easily, in atleast some part of question.
In that way bala's expl is much better and faster.
Earlier sol. that was posted can be used by ppl who have encountered this type of question for the first time to make understanding.

THANKS!! I agree with ontopofit but they both helped me understand the answer

excellent explanation! =)

Stuart Kovinsky wrote:

ieeyorei wrote:oops.
yes it's 7^(12x+3) + 3

however, i'm not sure how it is decided that the units would be 3 rather than, 7,9, or 1.

I think the explanation given is probably easier to understand than suresh's (which is correct, but a lot more complicated than we need), so let's reexamine that one.

For powers of 7, the units digit goes through a 4 part cycle:

7

9

3

1

So, to answer the question, we need to figure out where in the cycle we are.

Using our multiplication of exponent rule, we can change

7^(12x+3)

to

(7^12x) * (7^3)

Now let's look at each piece.

7^3 is easy.. that's the 3rd part of the cycle, so units digit will be "3".

7^12x seems more complicated, but it really isn't. We can rewrite 12x as 4*3*x, which is definitely going to be a multiple of 4. Therefore, no matter the value of x, 12x will always hit the 4 spot in the cycle. Accordingly, 7^12x will have a units digit of "1".

So, we're multiplying a units digit of 3 by a units digit of 1, giving us a units digit of 3.

One more step: now we're adding 3 to the total, giving us a units digit of 6.

When we divide a number with a units digit of 6 by 5, we'll end up with a remainder of 1... choose (B).