If x is a positive integer, what is the remainder

[laura.oppenheimer]

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

0,1,2,3,4

I'm struggling with how to do this problem if you don't instinctively know to look for a patter in base 7 numbers. Is there another way to approach this other than realizing the base 7 pattern?

[jnelson0612]

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

0,1,2,3,4

I'm struggling with how to do this problem if you don't instinctively know to look for a patter in base 7 numbers. Is there another way to approach this other than realizing the base 7 pattern?

Good question, Laura. Yes, you have to think about patterns when you see something like this. Consider that since x must be positive, the possibilities for this number from smallest on up include
(7^15) + 3
(7^27) + 3
(7^39) + 3

These are HUGE numbers. There is no way that we can figure out the number itself, and the GMAT knows that. Thus, there must be a pattern that we can exploit. I suspect patterns anytime I see a number with a large integer and an accompanying question that has something to do with the final digit.

[lyl9021]

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

0,1,2,3,4

first

the remainder of 1 divided by 5 is 1
the remainder of 2 divided by 5 is 2
the remainder of 3 divided by 5 is 3
the remainder of 4 divided by 5 is 4
the remainder of 5 divided by 5 is 0
the remainder of 6 divided by 5 is 1
the remainder of 7 divided by 5 is 2
the remainder of 8 divided by 5 is 3
the remainder of 9 divided by 5 is 4
the remainder of 10 divided by 5 is 0(it return every 5 numbers)
it would be use at second step


second
7^0=1 7^1=7 7^2=?9 7^3=?3
7^4=?1 7^5=7 7^6=?9 7^7=?3
so
it return every 4 number

12x+3=4*(3x)+3
it means the digit number of 7^(12x+3) is equal to 7^3 (it is 3)
and
7^(12x+3) + 3=?6
(explanation is from first step)

and the remainder is 1

my email is [redacted]
if you have any further problem on this math problem you can connect to me.
my first language is not English . So it would be some syntax error. sorry~

[tim]

thanks for your explanation. sorry i had to remove your email address, as personal email addresses are not appropriate to post here..

[alex.angola]

Hi everyone,

While i understand the last digit portion of the explanation, I don't quite get how you know 7^12x+3 is equivalent to 7^. If anyone can help with that portion, it would be great.

Thank you.

[RonPurewal]

Hi everyone,

While i understand the last digit portion of the explanation, I don't quite get how you know 7^12x+3 is equivalent to 7^. If anyone can help with that portion, it would be great.

Thank you.

Hi,
It appears you're missing some characters in your post. (Your second expression is 7 to the ... nothing at all.)

Also, the original moderator responses should have noted this, but we need the original source of this problem. If the original source is not posted, we'll have to lock or delete the thread.
Thanks.

[alex.angola]

Hi,

Sorry about that, I did forget a piece. My post should have read:

Hi everyone,

While i understand the last digit portion of the explanation, I don't quite get how you know 7^12x+3 is equivalent to 7^3. If anyone can help with that portion, it would be great.

Not sure what the original source of the problem is but I noticed it on my second CAT exam from Manhanttan GMAT.

Thank you.

[RonPurewal]

Hi,

Sorry about that, I did forget a piece. My post should have read:

Hi everyone,

While i understand the last digit portion of the explanation, I don't quite get how you know 7^12x+3 is equivalent to 7^3. If anyone can help with that portion, it would be great.

Not sure what the original source of the problem is but I noticed it on my second CAT exam from Manhanttan GMAT.

Thank you.

As mentioned earlier in the thread, you have to look at the series of powers of 7 until you notice a pattern in the units digits. (To know the remainder when you divide by 5, all you need to know is the units digit.)

so...
7^1 is 7
7^2 is 49, which ends with 9.
7^3 is 7 x (ends with 9), which ends with 3 (since 7 x 9 = 63).
7^4 is 7 x (ends with 3), which ends with 1 (since 7 x 3 = 21).
7^5 is 7 x (ends with 1) = ends with 7, so we're back to the starting point of the pattern.
So, the units digits are 7, 9, 3, 1, 7, 9, 3, 1, etc. forever.
In other words, if you move forward in this pattern by 4 spots, or 8 spots, or any multiple of 4 spots, then you have the same units digit.

The last key is to realize that 12x is a multiple of 4. That's why you can ignore it.