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Math Expert V
Joined: 02 Sep 2009
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In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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15 00:00

Difficulty:   55% (hard)

Question Stats: 60% (01:44) correct 40% (01:54) wrong based on 222 sessions

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In the product 2^19*9^13*5^24, what is the digit in the 20th place to the left of the decimal point?

A. 0
B. 2
C. 4
D. 5
E. 9

Kudos for a correct solution.

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Math Expert V
Joined: 02 Sep 2009
Posts: 58427
Re: In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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4
5
Bunuel wrote:
In the product 2^19*9^13*5^24, what is the digit in the 20th place to the left of the decimal point?

A. 0
B. 2
C. 4
D. 5
E. 9

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

When you're dealing with massive numbers created by the multiplication of exponents, 2s and 5s are your best friends; anywhere you can pair a 2 and a 5, you create a 10.

In this case, since you have 2^19 and 5^24, you'll have 19 pairings of 2 and 5, so you know that this product will end in 19 zeroes.

Having found that, that leaves you with 5^5*9^13. And since 5 multiplied by any odd number ends in a 5, you know that the next digit (working from the right hand side of this number) has to be a 5. Since the problem only asks about the 20th digit, you don't have to go any further with what would then be some pretty complicated math: the answer is 5.
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Math Expert V
Joined: 02 Aug 2009
Posts: 7960
Re: In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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1
1
ans D..5...
2^19*9^13*5^24= 2^19*5^19*9^13*5^5
as can be seen there are 19 zeroes and we basically have to find last digit of 9^13*5^5, which has to be 5
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Re: In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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Hi,

I cannot understand the explanation...

I understand up to this point: 10^19 * 5^5 * 9^13

I understand that 10^19 will have 20 numbers, starting with 1 and ending in 0.
I understand that 5^5 will end in 5 and that 9^13 will end with 9.

What I don't understand is how do we know that the 20th digit is the last digit?
What it the 20th digit is in the middle of the number?

So, in the end, we know that the multiplication of these three numbers looks like this:
10^19 = 1............0
5^5 = ......5
9^13 = .......9

What I don't know is how do we know where the 20th digit lies? I understand that 5^5*9^13 ends with 5, but how do we know what the 20th digit is if we multiply the result of 5^5*9^13 with 10^19?

If we were to find the last digit I would say that ok, it is possible, however, in this case it would end with 0, because the last digit of 10^19 * 5^5 should be zero (found by multiplying the last digits of each number: 0*5) and the last digit of that number multiplied by 9^13 (again multiplying the last digits: 0*9) would again be zero.

Some help...? Math Expert V
Joined: 02 Aug 2009
Posts: 7960
Re: In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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pacifist85 wrote:
Hi,

I cannot understand the explanation...

I understand up to this point: 10^19 * 5^5 * 9^13

I understand that 10^19 will have 20 numbers, starting with 1 and ending in 0.
I understand that 5^5 will end in 5 and that 9^13 will end with 9.

What I don't understand is how do we know that the 20th digit is the last digit?
What it the 20th digit is in the middle of the number?

So, in the end, we know that the multiplication of these three numbers looks like this:
10^19 = 1............0
5^5 = ......5
9^13 = .......9

What I don't know is how do we know where the 20th digit lies? I understand that 5^5*9^13 ends with 5, but how do we know what the 20th digit is if we multiply the result of 5^5*9^13 with 10^19?

If we were to find the last digit I would say that ok, it is possible, however, in this case it would end with 0, because the last digit of 10^19 * 5^5 should be zero (found by multiplying the last digits of each number: 0*5) and the last digit of that number multiplied by 9^13 (again multiplying the last digits: 0*9) would again be zero.

Some help...? hi pacifist,
what is the 20th digit from right in 10^19...
it will be 1 as 10^19=1000(19 times 0)...
now whatever you multiply with 10^19 the answer will be multiplication of those numbers followed by 19 zeroes,..
so last digit of multiple will be the 20th digit from right..
example in this case..
5^5*9^13*100000..(19 zeroes).... so xy... 50000(19 zeroes)... so 5 is the 20th digit from right
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In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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1
chetan2u wrote:
pacifist85 wrote:
Hi,

I cannot understand the explanation...

I understand up to this point: 10^19 * 5^5 * 9^13

I understand that 10^19 will have 20 numbers, starting with 1 and ending in 0.
I understand that 5^5 will end in 5 and that 9^13 will end with 9.

What I don't understand is how do we know that the 20th digit is the last digit?
What it the 20th digit is in the middle of the number?

So, in the end, we know that the multiplication of these three numbers looks like this:
10^19 = 1............0
5^5 = ......5
9^13 = .......9

What I don't know is how do we know where the 20th digit lies? I understand that 5^5*9^13 ends with 5, but how do we know what the 20th digit is if we multiply the result of 5^5*9^13 with 10^19?

If we were to find the last digit I would say that ok, it is possible, however, in this case it would end with 0, because the last digit of 10^19 * 5^5 should be zero (found by multiplying the last digits of each number: 0*5) and the last digit of that number multiplied by 9^13 (again multiplying the last digits: 0*9) would again be zero.

Some help...? hi pacifist,
what is the 20th digit from right in 10^19...
it will be 1 as 10^19=1000(19 times 0)...
now whatever you multiply with 10^19 the answer will be multiplication of those numbers followed by 19 zeroes,..
so last digit of multiple will be the 20th digit from right..
example in this case..
5^5*9^13*100000..(19 zeroes).... so xy... 50000(19 zeroes)... so 5 is the 20th digit from right

Which means that the answer in the end will be this one:
....(4)50000000000000000000. So, 5 taking the 20th place from the left (which I decided not to read) followed by 19 zeros.

Right?? Did I finally get it...?
Math Expert V
Joined: 02 Aug 2009
Posts: 7960
In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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pacifist85 wrote:
chetan2u wrote:
pacifist85 wrote:
Hi,

I cannot understand the explanation...

I understand up to this point: 10^19 * 5^5 * 9^13

I understand that 10^19 will have 20 numbers, starting with 1 and ending in 0.
I understand that 5^5 will end in 5 and that 9^13 will end with 9.

What I don't understand is how do we know that the 20th digit is the last digit?
What it the 20th digit is in the middle of the number?

So, in the end, we know that the multiplication of these three numbers looks like this:
10^19 = 1............0
5^5 = ......5
9^13 = .......9

What I don't know is how do we know where the 20th digit lies? I understand that 5^5*9^13 ends with 5, but how do we know what the 20th digit is if we multiply the result of 5^5*9^13 with 10^19?

If we were to find the last digit I would say that ok, it is possible, however, in this case it would end with 0, because the last digit of 10^19 * 5^5 should be zero (found by multiplying the last digits of each number: 0*5) and the last digit of that number multiplied by 9^13 (again multiplying the last digits: 0*9) would again be zero.

Some help...? hi pacifist,
what is the 20th digit from right in 10^19...
it will be 1 as 10^19=1000(19 times 0)...
now whatever you multiply with 10^19 the answer will be multiplication of those numbers followed by 19 zeroes,..
so last digit of multiple will be the 20th digit from right..
example in this case..
5^5*9^13*100000..(19 zeroes).... so xy... 50000(19 zeroes)... so 5 is the 20th digit from right

Which means that the answer in the end will be this one:
....(4)50000000000000000000. So, 5 taking the 20th place from the left (which I decided not to read) followed by 19 zeros.

Right?? Did I finally get it...?

yeah you are absolutely correct in your understanding.. however 5 will be the 20th digit from right..
19 zeroes and 20th is 5
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Joined: 07 Apr 2014
Posts: 402
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Re: In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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Ok, so there is a different perception of what to the left or to the right mean, hopefully! Othrwise I am still missing sth...

Well, the end point is that, if you start from the last digit of the number and count 20 numbers backwards, you will end up on the number 5!
Math Expert V
Joined: 02 Aug 2009
Posts: 7960
Re: In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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pacifist85 wrote:
Ok, so there is a different perception of what to the left or to the right mean, hopefully! Othrwise I am still missing sth...

Well, the end point is that, if you start from the last digit of the number and count 20 numbers backwards, you will end up on the number 5!

yeah, you are right 'left' and 'right ' makes a difference..
just an example.. in 10^19.. the 20th digit from right is 1 and from left is 0.....
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Re: In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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Bunuel wrote:
In the product 2^19*9^13*5^24, what is the digit in the 20th place to the left of the decimal point?

A. 0
B. 2
C. 4
D. 5
E. 9

Re-expressing 5^24 as 5^19 x 5^5, we can simplify the given expression:

2^19*9^13*5^24 = 2^19 x 5^19 x 9^13 x 5^5

Now, we combine 2^19 with 5^19, obtaining:

9^13 x 5^5 x 10^19

Recall that any number of the form m x 10^n (where m and n are positive integers) is the number m followed by n zeros. Since 9 raised to any power will always be odd, and since 5^5 will always end in a 5, 9^13 x 5^5 will end in a 5. Thus, the product 9^13 x 5^5 x 10^19 is a number ends with 19 zeros with the first nonzero digit to the left of these 19 zeros being a 5, which is also the 20th place to the the left of the decimal point.

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Re: In the product 2^19*9^13*5^24, what is the digit in the 20th place to  [#permalink]

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