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If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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Difficulty:   5% (low)

Question Stats: 85% (01:24) correct 15% (01:30) wrong based on 166 sessions

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If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

1
4
5
7
8

The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance

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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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1
Answer is 4 :

Cycle for 4^a is 4 or 6 : 4 16 64 256 1024 4096
cycle for 9^b is 1 or 9 : 9 81 729 6561

so possible units digit of xy = 4 or 6

Originally posted by MisterQ on 10 Oct 2011, 11:22.
Last edited by MisterQ on 10 Oct 2011, 11:27, edited 1 time in total.
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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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shrive555 wrote:
If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

1
4
5
7
8

The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance

x=4^a
a is +ve integer.
a=1; x=4
a=2; x=4^2=16
a=3; x=4^3=64
a^4=4*4=6

You see; the units digit of 4^x is either 4 OR 6.
Likewise;
units digit of 9^x is either 9 OR 1.

Possible units digit;
4*9=36
4*1=4
6*9=54
6*1=6

6 is not in the options, but 4 is.

Ans: "B"
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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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got it.. ..Thanks guys !!
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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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x=4^a
y=9^b

xy =(4^a)*(9^b)

when a=1,b=1 , xy 's unit digit =i.e unit's digit of (4*9= 36) i.e 6

a=1,b=2, xy 's unit digit = i.e units digit of (4*81) i.e 4

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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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Cyclisity of 4^n is 4,6,4,6
9^n is 9,1,9,1

given the question, we can have

9*4=36, 6
9*6=54, 4
1*4=4, 4
1*6=6, 6

therefore 4
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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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shrive555 wrote:
If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

1
4
5
7
8

The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance

4^a will end in either 4 or 6 .
9^b will always end in either 9 or1 .

So the possible unit digit of xy can be either of one below :

4*9 =6 or 4*1 = 4 OR 6*9 = 4 or 6*1 = 6.

So , as you can see the unit digit can either be 6 or 4.

We have just 4 in the choices.Hence , option (B) 4 is the correct option.
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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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shrive555 wrote:
If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

1
4
5
7
8

The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance

Another thing: $$x = 4^a$$ and $$y = 9^b$$
When you multiply, $$x*y = 4^a * 9^b$$
You cannot bring the powers together. Only if the bases are the same, then you can add the powers. e.g.
$$x = 4^a$$ and $$y = 4^b$$
When you multiply, $$x*y = 4^{a+b}$$

Or when the powers are the same, then you can multiply the bases together
e.g.
$$x = 4^a$$ and $$y = 9^a$$
When you multiply, $$x*y = 36^a$$
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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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VeritasPrepKarishma wrote:
shrive555 wrote:
If a and b are positive integers and x = 4^a and y = 9^b, which of the following is a possible units digit of xy?

1
4
5
7
8

The cyclicity of 4 is 6 when power is even and 2 when power is odd, similarly cyclicity of 9 is 2 when power is odd and 1 when power is even. since a & b are in the power, i multiply xy = 45^a+b , Now integer 45 is ending with 45 so unit digit of 45 will be base digit 5. but original answer is different Any thoughts on this question , thanks in advance

Another thing: $$x = 4^a$$ and $$y = 9^b$$
When you multiply, $$x*y = 4^a * 9^b$$
You cannot bring the powers together. Only if the bases are the same, then you can add the powers. e.g.
$$x = 4^a$$ and $$y = 4^b$$
When you multiply, $$x*y = 4^{a+b}$$

Or when the powers are the same, then you can multiply the bases together
e.g.
$$x = 4^a$$ and $$y = 9^a$$
When you multiply, $$x*y = 36^a$$

Thanks for pointing out!!
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Re: If a and b are positive integers than x=4^a & y=9^b  [#permalink]

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_________________ Re: If a and b are positive integers than x=4^a & y=9^b   [#permalink] 02 Mar 2018, 11:36
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