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Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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Jamboree and GMAT Club Contest Starts

QUESTION #14:

If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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We are given that N = ( 1436)^A*(1054)^B

Note that the unit's digit of N will depend on unit digits ( 1436)^A and (1054)^B which further depends on unit's digit of individual numbers 6^A and 4^B
Now since A is positive unit's digit of 6^A will be 6 irrespective of value of A.
and since B is positive, unit's digit of 4^B will be 4 or 6 depending on value of B.

So if we know B we can solve this question.
1) is clearly insufficient as we can different values of B and hence different values of unit's digit.

2) is sufficient to solve this question as we just need value of B.

Ans. B
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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = (1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

(1436)^A -- units digit will always be 6 for all positive powers.
(1054)^B -- units digit will be 4 for odd powers
and units digit will be 6 for even powers

1. A and B can take values from 1-5 .
Unit digit for N will depend on units digit of (1436)^A and (1054)^B.
If A and B are odd here , the units digit of N will be 6* 4 = 24
Therefore units digit will be 4
But if A and B are even , the units digit will be 6*6= 36
Therefore units digit will be 6

Not sufficient.

2. B=2
units digit of (1054)^B will be 6 .
Therefore units digit of N will be 6

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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To find units digit of N, we must know value of A & B.
1436^A, its unit digit will always be 6 for positive integers
1) A+B = 6, from this we cannot find out the value of both the variables so, INSUFFICIENT
2) B= 2, from this, we know that unit digit of 1054^2 will be 6 & 1436^A will also be 6 no matter what will be the value of A is, unless it is a positive integer. So 6x6=36, mean 6 unit digit. B is Sufficient
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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

the unit digit of N = unit digit of {unit digit of ( 1436)^A * unit digit of (1054)^B}
Now, unit digit of ( 1436)^A = 6, irrespective of A is odd or even.
but, unit digit of (1054)^B = 4 when B id odd and unit digit of (1054)^B = 6 when B is even.

1. A+B = 6.
a. A = 1, B = 5. B is odd.
b. A = 2, B = 4. B is even. Not Sufficient.

2. B = 2, B is even. Sufficient.

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

this question can be rewritten as: What is the units digit of the (1054)^B? or, even more precise - is B even or odd???
Why? Because to find the units digit of N, we need to find the units digit of ( 1436)^A, and the units digit of (1054)^B.
Regardless of what A is, the units digit of ( 1436)^A will always be 6.

The pattern for 4^x is:
if x is odd - the units digit will be 4
if x is even - the units digit is 6.

1) A+B = 6, well A can be 1 and B can be 5. This will yield a units digit 4 for the number N or
A can be 2 and B can be 4, which will yield a units digit 6 for the number N.
As such, statement 1 is insufficient.

2) B=2. we know that B is even, we thus know the units digit of (1054)^B, and thus, we can find the units digit of N.

B - statement 2 alone is sufficient is the answer.
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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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QUESTION #14:

If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

From Question, 6 takes only 6 value in unit digit for any value of A and for 4, it gives 4 for odd values of B & it gives 6 for even value of B.

1. A+B=6,
> B = 6 - A; B can take any values, i.e., 6,5,4,3,2,1,0 . Hence not sufficient.

2. B = 2, Sufficient
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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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Stmt 1: A+ B =6
possible values for A = 1, 2 ,3 ,4, 5
possible values for B = 1, 2 ,3 ,4, 5

(1436)^A: For all values of A, the units digit will be 6.
(1054)^B: Unit digit can vary based on the value of B.

Therefore, insufficient.

Stmt 2: B = 2

1436)^A: For all values of A, the units digit will be 6.
(1054)^B: For B =2, unit digit will be 6. Therefore, the unit digit of 1436)^A * (1054)^B will be 6.

Sufficient.

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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6^1 = 6, 6^2 = 36, 6^3 = 216 .... So, for any power of 6 will have units digit of 6
4^1 = 4, 4^2 = 16, 4^3 = 64, 4^4 = 256 ... odd powers of 4 will have units digit as 4, even powers of 4 will have units digit as 6
for any positive integer A, (1436)^A will have units digit 6
In (1054)^B = will have units digit of 4, If B is odd
=will have units digit of 6, If B is even

Statement (1) => A + B = 6
if A = 1, B = 5
N = ( 1436)^A*(1054)^B = ( 1436)^1*(1054)^5 = (units digit of 6)*(units digit of 4) = units digit of 4
if A = 2, B = 4
N = ( 1436)^A*(1054)^B = ( 1436)^2*(1054)^4 = (units digit of 6)*(units digit of 6) = units digit of 6
So, Not sufficient

Statement (2) => B = 2
N = ( 1436)^A*(1054)^B = ( 1436)^A*(1054)^2 = (units digit of 6)*(units digit of 6) = units digit of 6
Sufficient

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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N = ( 1436)^A*(1054)^B. Units digit of N = ?

Since 1436 ends with 6, units digit of (1436)^A always ends with 6.
Units digit of (1054)^B can be 4 or 6. So, we need the value of B to find the units digit of N.

St1: A + B = 6 --> Does not provide the value of B. Hence not sufficient.

St2: B = 2 --> Units digit of (1054)^2 ends with 6.
Since units digit of both (1436)^A and (1054)^B is 6, units digit of N = 6.
Statement 2 alone is sufficient.

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

Question: What is the units digit of N = ( 1436)^A*(1054)^B ?
-> What is the units digit of 6^A * 4^B?
-> What is the units digit of 6 * 4^B (6 to the power any integer results in a number with the units digit 6)
-> Is B odd or even? ( 4^odd integer = units digit 4, 4^even integer = units digit 6)

Statement 1: A + B = 6
B could be even or odd depending on A
Therefore, INSUFFICIENT!

Statement 2: B = 2
B is even
Therefore, SUFFICIENT!

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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Please refer to the picture for the solution.

Attachment: IMAG0112.jpg [ 1.77 MiB | Viewed 2404 times ]

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

N = ( 1436)^A*(1054)^B

Units digit of ( 1436)^A will always be 6 irrespective of value of A(since it is >=1)
Units digit of (1054)^B will be 4 if B=odd and 6 if B=even.

So basically the question here is if B is even or odd.

Stmt1: (1) A + B = 6
If A=1 B=5, B is odd
If A=2 B=4, B is even
-> Not conclusive -> Not sufficient

Stmt2:
(2) B = 2 -> B is even -> Sufficient

Hence Ans:B
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I think attachments are not hidden from others. I can see the answer of saunakdey, who has posted an attachment.
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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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B because units digit of 1436 will always end in 6. Units digit of 1054 can end in a 6 or a 4.

Statement 1: A+B=6 -----> B=6-A---> Not Sufficient
Statement 2: B=2 so 1054 will end in a 6. Units digit of N=xxxxxxx6
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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

We need to find the unit digit. so immidiately see the unit digit of nummber to be multiplied.
In case of 1436 - its unit digit is 6.
In case of 1054 - its unit digit is 4.

Now lets see what is the resultant unit digit when 4 or 6 is multiplied even or odd number of times with itself.

Check for 6:
Remember, whenever unit digit of a number is 0,1,5 or 6, the unit digit would remain same irrespective of the powers.
(6)^2 = 6*6 = 36 (unit digit - 6)
(6)^3 = 6*6*6 = 216 (unit digit - 6)
So power can be even or odd but unit digit would reain same.

Check for 4:
In case of 4, if the power is even then unit digit is 6 else if the power is odd then unit digit is 4.
(4)^2 = 4*4 = 16 (unit digit - 6)
(4)^3 = 4*4*4 = 64 (unit digit - 4)
(4)^4 = 4*4*4*4 = 256 (unit digit - 6)
(4)^5 = 4*4*4*4*4 = 1024 (unit digit - 4)

lets see the statements individually to check sufficiency.
(1) A + B = 6

in case of 1436 - its unit digit would remain 6. Independent of the value of "A".
in case of 1054 - its unit digit can be 4 or 6 because it depends on whether "B" is even or odd.

SO here the final unit digit can be 4 OR 6, hence INSUFFICIENT.

(2) B = 2

in case of 1436 - its unit digit would remain 6. Independent of the value of "A".
in case of 1054 - its unit digit will be 6 because "B" is even.
so, here, final unit digit would be "6". SUFFICIENT.

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

The units digit of 1436 will only be 1 if A = 0. But we know that A is positive, so 1436^A will always be 6, regardless of the value of A.
Similarly 1054 will never be 1 since B is positive. But the units digit of 1054 is 4 and 4 has a cyclicity of 2 - 4, 6.

In short if B is odd, then the units digit of N is 6*4=4, but if B is even then the units digit of N is 6*6=6.

Hence we only need the value of B.

1. Not Sufficient.
B could be anything between 1 to 5

2. Sufficient
We have the exact value of B.

Hence the units digit of N is 6*6=6

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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The unit digit of N depends on the unit digit of "(1436)^A" multiples the unit digit of (1054)^B

- (1436)^A always ends in 6 if A is positive integer (6^1, 6^2, ....)
- (1054)^B ends in 4 if B is odd integer (4^1, 4^3....) or in 6 if B is even integer (4^2, 4^4....).

So N ending in 4 (6*4) or in 6 (6*6) depends on B is odd or even.

(1)A+B=6 => B can be odd (1,3,5) or even (2,4) => N ends in 4 or 6 =>NOT SUFFICIENT

(2) B=2 => N ends in 6 => SUFFICIENT

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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

No matter what is A, the last digit of the first number is 6 and depending on the B the last digit of the second number is 4 or 6. So we need to know about B. Since 4 has cyclicity of 2 at least we should know if B is even or odd.

(1) A + B = 6. We can not find if B is even or odd or exact B.

(2) B = 2. Yes. B is 2 or even. So the last digit of 1054 is 6. The last digit of N is then 6*6=...6.
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Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and  [#permalink]

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If N = ( 1436)^A*(1054)^B. Where A and B are positive integers. What is the units digit of N?

(1) A + B = 6
(2) B = 2

Whatever be the value of A (>0 as per question), the unit digit value will be always 6. So the answer to this question depends on value of B.
1)Taking choice 1 A+B=6
B can take any values from 1 to 5. So this choice is not sufficient.

2)Taking choice 2 , B=2
Since B=2, the units digit of both will be 6*6= 6. So this is sufficient to answer the question. Re: Jamboree and GMAT Club Contest: If N = ( 1436)^A*(1054)^B. Where A and   [#permalink] 20 Nov 2015, 10:08

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