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# If x is a positive integer, what is the remainder when 7^(12

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If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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13 Oct 2009, 06:33
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If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4
[Reveal] Spoiler: OA

Last edited by Bunuel on 17 Oct 2013, 02:08, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Tuesday Q5 - Remainder [#permalink]

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13 Oct 2009, 07:53
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[7^( 12x + 3 ) + 3 ] can be rewritten as -

[(7^(12x) . 7^3) + 3 ]
[ (7^(4)(3x) . 7^3 ) +3 ]

now 7 cube is 343 the unit digit is '3',
and 7 raised to the power of 4 is 2401, unit digit is 1 and inturn raised to the power 3x will result with unit digit as 1,

so the equations boils down to [( 1 . 3 ) + 3]/5 which is 1
so the answer is B
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Re: Tuesday Q5 - Remainder [#permalink]

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13 Oct 2009, 11:20
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7^(12x+3) + 3 is divided by 5

for x = 1 >> R = 1

7^(15) + 3 ..cyclicity of 7 is 4 hence 15/4 ... R >> 3 corresponds to 7,9,3,1
hence 3+3 = 6 /5 R = 1 OA B
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Re: Tuesday Q5 - Remainder [#permalink]

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27 Oct 2009, 16:53
kirankp wrote:

and 7 raised to the power of 4 is 2401, unit digit is 1 and inturn raised to the power 3x will result with unit digit as 1,

I couldn't pick this one out, if unit digit is 1 and the power is 3x how can the resulting unit digit be 1 if x is unkown???

Any one there?????
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Re: Tuesday Q5 - Remainder [#permalink]

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31 Jan 2010, 06:32
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hogann wrote:
If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?
1. 0
2. 1
3. 2
4. 3
5. 4

Edit - Fixed Exponent

7^(12x+3)Mod 5 + 3 Mod 5
= (7 mod 5)^12x * (7 mod 5)^3 + 3
= (2 mod 5)^12x * (2^3 Mod 5) + 3
= (4 mod 5)^6x * (8 Mod 5) + 3
= (-1)^6x * (-2) + 3
= 1 * (-2) + 3
= 1

Therefore B!
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Re: Tuesday Q5 - Remainder [#permalink]

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13 Feb 2010, 00:24
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(7^(12x+3) + 3)/5

x is a positive integer, so we can put the value 1,2,3......

for x=1 x=2 or x=3
7^15 7^27 7^39
7 has the cycilicity of 4
15 mod 4, 27 mod 4 or 39 mod 4 leads to same result which is 7^3 will be unit digit = 343.

343+3/5 = gives 1 as remainder

B is the answer
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If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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06 Oct 2012, 09:55
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If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4

[Reveal] Spoiler:
Can someone explain how to solve this in a different way than the MGMAT CAT Test does?

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 712x+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

71

72

73

74

75

76

77

78

712x

712x+1

712x+2

712x+3

We can 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 712x would end in a 1, just like 74 or 78.
712x+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 712x+3 ends in a three, 712x+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.
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Re: (7 ^ 12x+3) + 3 divided by 5; what is the remainder? [#permalink]

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06 Oct 2012, 10:24
Honestly, I don't see another way.

With this specific question, you can also assume x=0 and get the correct result, although the question states that this is not allowed. But since this is a dangerous approach, where you have to know that it will lead to the same result as a positive integer, I don't recommend it.

But anyways: if x=0, then 7^3 + 3. 49*7+3=346, so remainder is 1.
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Re: (7 ^ 12x+3) + 3 divided by 5; what is the remainder? [#permalink]

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06 Oct 2012, 10:29
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I have put in a solution in the below format. Plz see if it helps.
Attachments

Remainder Problem.png [ 9.72 KiB | Viewed 7218 times ]

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Re: (7 ^ 12x+3) + 3 divided by 5; what is the remainder? [#permalink]

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06 Oct 2012, 10:37
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TheFerg wrote:
If x is a positive integer, what is the remainder when (7 ^ 12x+3) + 3 is divided by 5?

Can someone explain how to solve this in a different way than the MGMAT CAT Test does?

0
1
2
3
4

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 712x+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

71

72

73

74

75

76

77

78

712x

712x+1

712x+2

712x+3

We can 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 712x would end in a 1, just like 74 or 78.
712x+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 712x+3 ends in a three, 712x+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.

I guess it should be $$7^{12x+3}+3$$. The solution explains that the last digits of the powers of $$7$$ repeat in a cyclical manner: $$7, 9, 3, 1, 7, 9, 3, 1,...$$
Meaning, if the exponent is a multiple of 4 + 1 ($$M4+1$$ like 1, 5, 9,...), the last digit is $$7,$$ for a $$M4+2$$ (like 2, 6, 8,...) the last digit is $$9,$$ etc.

$$1$$ is also a $$M4 + 1$$, because $$1 = 4\cdot0 + 1.$$

In this question, it is obvious the answer does not depend on the value of $$x$$. But, I am going to tell you a secret: even if they say $$x$$ is positive, you can still plug in $$0$$ for $$x,$$ the answer will be the correct one. So, just look at $$7^3+3$$, and find the last digit, which is $$6,$$ so the remainder when dividing by $$5$$ is $$1.$$
They just state that $$x$$ is positive, meaning at least $$1,$$ so that you cannot raise $$7$$ to the $$12\cdot1+3=15$$th power.
Since $$12x$$ is a multiple of $$4,$$ $$\,\,12x + 3$$ is a $$M4+3.$$ In other words, only the remainder is important, you can ignore $$12x$$ or just take it as 0.

Answer B.
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Re: (7 ^ 12x+3) + 3 divided by 5; what is the remainder? [#permalink]

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06 Oct 2012, 10:41
Zinsch123 wrote:
Honestly, I don't see another way.

With this specific question, you can also assume x=0 and get the correct result, although the question states that this is not allowed. But since this is a dangerous approach, where you have to know that it will lead to the same result as a positive integer, I don't recommend it.

But anyways: if x=0, then 7^3 + 3. 49*7+3=346, so remainder is 1.

In this case, no danger to consider x = 0. The only restriction should be 12x + 3 non-negative. The powers of 7 have their last digit repeating cyclically, and for x = 0 the exponent is positive, so no problem.
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Re: (7 ^ 12x+3) + 3 divided by 5; what is the remainder? [#permalink]

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07 Oct 2012, 11:02
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Many good insights above. I'll add: there is actually a whole class of questions on the GMAT that ask either "what is the remainder when x is divided by 5" or "what is the remainder when x is divided by 10"? Both of these questions can be answered just by knowing the units digit of x. The reason why that works should be fairly clear if you do a few division problems (try finding the remainder for 7÷5, 37÷5, and 127÷5).

Once you understand that part of the problem, the question becomes "what is the units digit of (7 ^ 12x+3) + 3?" To figure that out, we first need to know the units digit of (7 ^ 12x+3), since the final +3 will be easy to deal with.

If you have no idea what to do at this point, I think the best thing is just to figure out what the units digit of 7^1 is, then of 7^2, then of 7^3, then of 7^4, etc. It's easy for the first two (7 and 9), but remember that to find the units digit of 7^3, you don't actually need to figure out 49*7. You just need to figure out 9*7. 9*7 is 63 (and by the way, 40*7 is 280 - since that ends in a 0, it won't affect the units digit when we add it to 63 to figure out what 7^3 is). So the units digit of 7^3 is 3. And the units digit of 7^4 is therefore the same as the one of 3*7, or 1. The units digit of 7^5 is 7 again, because 1*7 is 7. Note that once you get to a units digit of 1, that's when the pattern starts repeating. Check out Pansi's chart above.

So assume x=1. The units digit of 7^15 is the same as 7^11, 7^7, and 7^3. That would be 3. Add 3 to that and you get 6. Divide by 5 and you get a remainder of 1.

Final note: 7^x isn't the only base with an interesting units digit pattern:
2^x goes 2-4-8-6
3^x goes 3-9-7-1
4^x goes 4-6
5^x is always 5
6^x is always 6
8^x goes 8-4-2-6
9^x goes 9-1
and by the way, 17^x goes 7-9-3-1 just like 7^x does. So does 27^x. You get the idea. No need to memorize these patterns, just understand how to figure them out.
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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05 Sep 2013, 11:41
Without having to keep writing out beyond the basic pattern of the unit digits of 7, which is 7,9,3,1..... how do you determine that 7^12X is multiple of 4 instead of 2 or 3 and use that respective digit? How do you determine that 7^15 unit digit is based on 7^3 instead of 7^5. I opted to solve by plugging in 1 for X and thought that the unit digit for 7^15 was 7 because I based it on 7^1 and 7^5. Thanks!!!
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Re: Tuesday Q5 - Remainder [#permalink]

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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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25 Oct 2013, 19:25
I plugged numbers into the formula and did calculation to the 15th power to determine units digits
7^(12x+3)+3 using x=1

1,2 - 7's = 49
3,4 - 7's = 49 ---> 1,2,3,4 7's = 49*49 ---> units digits = 1
5,6 - 7's = 49
7,8 - 7's = 49 ---> 5,6,7,8 7's = 49*49 ---> units digits = 1
9,10 - 7's = 49
11,12 - 7's = 49 ---> 9,10,11,12 - 7's = 49*49 ---> units digits = 1
13,14 - 7's = 49
15 - 7's = 7 ---> 13,14,15 - 7's = 49*7 ---> units digits = 3

1 * 1 * 1 * 3 = 3 + 3 = 6/5= 1
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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14 Nov 2013, 05:42
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hogann wrote:
If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4

Way I did it was:

Pick x=1.
7^15+3/5

Then split them

7^15/5 + 3/5

Then 7 = (5+2) ^15

So then every term will be divisible by 5 except 2^15.
The units digit on 2^15 is going to be 8. So 8/5 Remainder 3.

Now, addind 3+3 = 6.

6/5 gives remainder 1.

So answer is B

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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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27 Jan 2014, 05:41
hogann wrote:
If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4

Binomial method works well here.
First lets assume x=1

so 7^12x+3 + 3 / 5 what is the remainder?

7^15 + 3 /5

(5+2)^15 / 5, all terms will be divisible by 5 except 2^15

What is remainder of 2^15 / 5?

2^15 = 2* 4^7 = 2(5-1)^7

All terms will again be divisible by 5 except -1^7
Then -1*2 = -2 + 3 = 1

Hence we have a remainder of 1 when divided by 7

Answer is thus B

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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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03 Feb 2014, 15:20
The way i did it :
7^(12x+3) -- > 7^(12x) * 7^3
7 has a cyclicity of 4 so
- 12x/4 will always give us a remainder of 0 thus last digit is 1.
- 3/4 remainder 3 so the last digit is 3 .
So we know the last digit of 7^(12x+3) is 3*1 = 3
That last digit will be added to 3 since we have 7^(12x+3) + 3
so 3 +3 = 6
6/5 will give us a remainder of 1 so B.
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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14 Feb 2014, 15:43
Different approach:

Since x is a positive integer, assume x = 1.

7^(12x+3) + 3 = 7^(15) + 3 = (5+2)^15 + 3

The remainder of 5^15 / 5 will be 0, so that leaves us with 2^15 + 3

Cycle of 2^15 becomes easy to work with:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
[units digit of 2^x repeats]
...
Working through the cycle, we know that 2^15 = xx8.
So, the final number will be xx8 + 3 = xx1. xx1/5 will leave a remainder of 1.

Answer: B
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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21 Mar 2014, 05:50
jlgdr wrote:
hogann wrote:
If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4

Binomial method works well here.
First lets assume x=1

so 7^12x+3 + 3 / 5 what is the remainder?

7^15 + 3 /5

(5+2)^15 / 5, all terms will be divisible by 5 except 2^15

What is remainder of 2^15 / 5?

2^15 = 2* 4^7 = 2(5-1)^7

All terms will again be divisible by 5 except -1^7
Then -1*2 = -2 + 3 = 1

Hence we have a remainder of 1 when divided by 7

Answer is thus B

Cheers!
J

Just wondering is it possible to multiply both numerator and denominator by 2 in order to get 10 as denominator? That way we would only need to worry about the units digit of the expression

Bunuel, could you clarify whether this approach is valid?

Thanks
Cheers
J
Re: If x is a positive integer, what is the remainder when 7^(12   [#permalink] 21 Mar 2014, 05:50

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