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

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

(a) 0
(b) 2
(c) 5
(d) 13
(e) 17
[Reveal] Spoiler: OA
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Re: How many zeros does 135^19 x 42^17 end with? [#permalink]

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

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