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

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

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New post 07 Sep 2013, 03:56
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Question Stats:

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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
[Reveal] Spoiler: OA

Kudos [?]: 17 [0], given: 15

Manager
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Posts: 157

Kudos [?]: 40 [0], given: 559

Location: India
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Re: How many zeros does 135^19 x 42^17 end with? [#permalink]

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New post 07 Sep 2013, 05:45
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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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Kudos [?]: 40 [0], given: 559

Manager
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Re: How many zeros does 135^19 x 42^17 end with? [#permalink]

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New post 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)

So answer is 17

Kudos [?]: 87 [0], given: 72

Senior Manager
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Posts: 369

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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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New post 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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Kudos [?]: 91 [0], given: 349

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

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New post 03 Aug 2016, 13:30
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 \(2 \times 5\)

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

[Reveal] Spoiler:
(e) 17

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Kudos [?]: 25 [0], given: 51

How many zeros does 135^19 x 42^17 end with?   [#permalink] 03 Aug 2016, 13:30
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