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# How many zeros does 135^19 x 42^17 end with?

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Intern
Joined: 28 Jan 2011
Posts: 22
How many zeros does 135^19 x 42^17 end with?  [#permalink]

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07 Sep 2013, 03:56
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Difficulty:

25% (medium)

Question Stats:

71% (00:58) correct 29% (01:19) wrong based on 106 sessions

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How many zeros does 135^19 x 42^17 end with?

(a) 0
(b) 2
(c) 5
(d) 13
(e) 17
Manager
Joined: 20 Jul 2012
Posts: 140
Location: India
WE: Information Technology (Computer Software)
Re: How many zeros does 135^19 x 42^17 end with?  [#permalink]

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07 Sep 2013, 05:45
1
1
salsal wrote:
How many zeros does 135^19 x 42^17 end with?

(a) 0
(b) 2
(c) 5
(d) 13
(e) 17

135=5x3^3 this implies
135^19=5^19x(3^3)^19..
similarly 42=2x3x7
42^17=2^17x3^17x7^17

we know to get zero we need 2 and 5... and we have 19 5's and 17 2's ...thus we can get 17 10's
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Manager
Joined: 30 May 2013
Posts: 164
Location: India
Concentration: Entrepreneurship, General Management
GPA: 3.82
Re: How many zeros does 135^19 x 42^17 end with?  [#permalink]

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07 Sep 2013, 06:18
salsal wrote:
How many zeros does 135^19 x 42^17 end with?

(a) 0
(b) 2
(c) 5
(d) 13
(e) 17

To know the zeros we need to know how many 10's are there for this need to find number of 2s and 5s ( 2 * 5 = 10)

135 = (5 x 3 ^3)^19 So Number of 5s are = 19
42 = (2x7x3)^17 so number of 2s = 17

So tot number of 10s in this number is (17-> 2s x 17-> 5s)

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Joined: 02 Dec 2014
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Location: Russian Federation
Concentration: General Management, Economics
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Re: How many zeros does 135^19 x 42^17 end with?  [#permalink]

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16 Jul 2016, 07:28
5^19 always end on 5. 2^17 has a cycle of 4 (2,4,8,2 and again 2,4,8,2) Since 5 multiplied by each number from cycle of 2 has 0 in the end then the whole number will have 17s zero (since number of 2s is 17)
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Manager
Joined: 07 Jul 2016
Posts: 79
GPA: 4
How many zeros does 135^19 x 42^17 end with?  [#permalink]

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03 Aug 2016, 13:30
1
salsal wrote:
How many zeros does $$135^{19} \times 42^{17}$$ end with?

Take the prime factorisations of the two numbers.

$$135 = 5 \times 3^3\\ 42 = 2 \times 3 \times 7$$

$$135^{19} = 5^{19} \times {(3^3)}^{19}\\ 42^{17} = 2^{17} \times 3^{17} \times 7^{17}$$

The prime factorisation of 10 is $$2 \times 5$$

$$135^{19} \times 42^{17} = \color{blue}{(2^{17} \times 5^{17})} \times 5^{2} \times {(3^3)}^{19} \times 3^{17} \times 7^{17} =\\ \color{blue}{10^{17}} \times 5^{2} \times {(3^3)}^{19} \times 3^{17} \times 7^{17}$$

(e) 17

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Manager
Joined: 18 Jun 2018
Posts: 154
Re: How many zeros does 135^19 x 42^17 end with?  [#permalink]

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05 Sep 2018, 12:49
OA:E

$$135^{19} * 42^{17} = 3^{3*19}*5^{19}*2^{17}*3^{17}*7^{17}=2^{17}*3^{74}*5^{19}*7^{17}=10^{17}*3^{74}*5^2*7^{17}$$

$$10^{17}$$ will lead to $$17$$ zeros
Re: How many zeros does 135^19 x 42^17 end with? &nbs [#permalink] 05 Sep 2018, 12:49
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