After multiplying a positive integer A, which has n digits,

Question

After multiplying a positive integer A, which has n digits, by (n+2), we get a number with (n+1) digits, all of whose digits are (n+1). How many instances of A exist?

A. None
B. 1
C. 2
D. 8
E. 9

KarishmaB · Accepted Answer

The question seems convoluted but it's not. You have to take the first step in the right direction. The only definitive thing given here is that we get a number with (n+1) digits, all the digits being (n+1). What will such a number look like?

22
333
4444
55555 etc

We obtain this number by multiplying A with (n+2). This means that our number should be divisible by (n+2). Now, ask yourself:
Is 22 divisible by 3? No.
Is 333 divisible by 4? No 
We know that no odd number will be divisible by an even number. So we can ignore 333, 55555, 7777777 etc

Only consider even numbers:

Is 4444 divisible by 5? No

Is 666666 divisible by 7? Yes! Check: 666666/7 = 95238 (5 digit number). SO when you multiply 95238 by 7, you get 666666

Is 88888888 divisible by 9? No

Use divisibility rules to quickly rule out the numbers not divisible.

Answer (B)

Narenn · Answer

Constraint 1) when we put a sequence of multiples of (n+2) atleast one multiple should have its unit digit same as that of (n+1)
Constraint 2) N+1 can not greater than 9 since it is a single digit.
Constraint 2) N can not be 0

n(n+1)(n+2)
8---9----10------ Units digit zero always. so out
7---8----9-------- 9X2=18 so 2323232X9 = .......2988 or 2222222 x 9 = ......98 out
6---7----8-------- units digits 8,6,4,2,4,8,6,4,2,0,8,6.... No 7 so out
5---6----7-------- 7x8=56 so 83838 x 7 = .......5866 or 88888 x 7 = ....216 out
4---5----6-------- units digits 6,2,8,4,0,6,2,8,4,0...... No 5 so out
3---4----5-------- 5,0,5,0 out
2---3----4-------- 4,8,2,6,0,4.... out
1---2----3-------- only possible pair is 3 x 4 = 12 so out
 
Bunuel, Can you Pls help?

Narenn · Answer

You said correctly Karishma.
It is important to take first step in right direction.

Regards,

Abhijit

vikgupta07 · Answer

Hi Karishma,

How did you reach to 666666 specifically?

Also, the qualifying numbers in the problem could be 
11,22,33,44,55,66,77,88,99
111,222,333,444,555,666,777,888,999
1111....
11111....
111111...
and on

How do we narrow down this in a short span of time? Sorry, I could not understand this.

Thanks,
Vikram

AdityaHongunti · Answer

After multiplying a positive integer A, which has n digits, by (n+2), we get a number with (n+1) digits, all of whose digits are (n+1). How many instances of A exist?

Read the question carefully:

You say 11 could be one of the numbers- here's how it cannot-

11 is the product of A with n number of digits and (n+2).
Now to get 11 we have to have number of digits to be 0. Only then can the number be (n+1=0+1) and as the question tells us that all the numbers are all same then in no way can you get 11 a 2 digit number of n =0 as the maximum the product could be is of 1 digit.

Posted from my mobile device

amascarenhas · Answer

Hello,
I do not understand the question and the answers.
I cannot see written in the text that all digits of A are the same.

let's say that A = 4,5,6,7, 8 or 9 in this case n=1. If I multiply A by 3 (n+2) I will have a number with 2 digits.

If I choose a number with two digits: A= 25, 26, 27,..., 99 then A*4 will have 3 digits and so on..

So what didn't I understand?

AdityaHongunti · Answer

You may have been right about not understanding the question.

You got the first 2 parts right.

Third part- we get a number with (n+1) digits, all of whose digits are (n+1).

This means that product of A and n+2 has every digit same.
As in A= 95238 >>number of digits (n)= 5

Therefore A*(5+2) = 666666

The answer has 6 ie.(n+1) digits . And each digit in the answer is of the form (n+1)

So basically if you consider 
Say A=23 >> n=2
A* (n+2) = (n+1)*(n+1)*(n+1)

Here the answer has three digits ie. (n+1) digit with each digit as (n+1)

Now the above example actually does not exist as there is only one such number and that is 666666

Posted from my mobile device

KarishmaB · Answer

Read this: "... we get a number with (n+1) digits, all of whose digits are (n+1)"

The number has (n+1) digits. Each digit is (n+1).
So the possible numbers are:
1 digit number with each digit 1 i.e. 1
2 digit number with each digit 2 i.e. 22
3 digit number with each digit 3 i.e. 333
and so on...

vikgupta07 · Answer

After multiplying a positive integer A, which has n digits, by (n+2), we get a number with (n+1) digits, all of whose digits are (n+1). How many instances of A exist?

Read the question carefully:

You say 11 could be one of the numbers- here's how it cannot-

11 is the product of A with n number of digits and (n+2).
Now to get 11 we have to have number of digits to be 0. Only then can the number be (n+1=0+1) and as the question tells us that all the numbers are all same then in no way can you get 11 a 2 digit number of n =0 as the maximum the product could be is of 1 digit.

Posted from my mobile device

Thank you!

Your comment did somehow helped me solve my doubt. I got it sorted out.

vikgupta07 · Answer

Hi,

Let me try and explain

1. A=Positive Integer  , Let's say 5
2. Multiplication Factor:  , From 1, it's 5+2=7
3. Resultant Number: 5*7=35, which has   digits, according to the question, but all digits are not  , The required number, according to this case should have been 66

The only numbers that would satisfy the problem are: -

1
22
333
4444
55555
666666
7777777

and so on.

If this is the resultant number, then if it has   digits, it must have been divisible by  , FROM 2 Above. Do the quick math as to what number is divisible by

from the qualifying set above, 666666/7=95238 is a candidate.

This A should satisfy all boundaries of the problem. therefore instances=1

Hope this helps.

Thanks,
VIKRAM

fskilnik · Answer

\left. \matrix{
 A = \,\,\underbrace {\underline {} \,\,\underline {} \,\,\underline {} \,\, \ldots \,\,\underline {} }_{n\,\,{m{digits}}}\,\,\,\, \ge 1\,\,{\mathop{m int}} \hfill \cr 
 A \cdot \left( {n + 2} ight) = \underbrace {\underline {n + 1} \,\,\underline {n + 1} \,\,\underline {n + 1} \,\, \ldots \,\,\underline {n + 1} }_{n + 1\,\,{m{digits}}}\,\,\,\, \hfill \cr} ight\}\,\,\,\,\,\,\,\,?\,\,\,\, = \,\,\,\,\# \,\,A\,\,\,{m{possible}}

This is a typical  organized manual work technique  exercise!

\left\{ \matrix{
 n = 1\,\,\,\,\, \Rightarrow \,\,\,\,\underline {} \,\, \cdot \,\,\,\left( {1 + 2} ight) = \underline 2 \,\,\underline 2 \,\,\,\,,\,\,\,\,{m{impossible}}\,\,\,\,\,\left( {{{22} \over 3} 
e {\mathop{m int}} } ight) \hfill \cr 
 n = 2\,\,\,\,\, \Rightarrow \,\,\,\,\underline {} \,\underline {} \,\, \cdot \,\,\,\left( {2 + 2} ight) = \underline 3 \,\,\underline 3 \,\,\underline 3 \,\,\,\,\,,\,\,\,\,{m{impossible}}\,\,\,\,\,\left( {{{{m{odd}}} \over {{m{even}}}} 
e {\mathop{m int}} } ight) \hfill \cr 
 n = 3\,\,\,\,\, \Rightarrow \,\,\,\,\underline {} \,\underline {} \,\underline {} \,\, \cdot \,\,\,\left( {3 + 2} ight) = \underline 4 \,\,\underline 4 \,\,\underline 4 \,\,\underline 4 \,\,\,\,\,\,\,,\,\,\,\,{m{impossible}}\,\,\,\,\,\left( {{{4444} \over 5} 
e {\mathop{m int}} } ight) \hfill \cr 
 n = 4\,\,\,\,\, \Rightarrow \,\,\,{m{impossible}}\,\,\,\,\,\left( {{m{idem}}\,\,n = 2} ight) \hfill \cr 
 n = 5\,\,\,\,\, \Rightarrow \,\,\,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\, \cdot \,\,\,\left( {5 + 2} ight) = \underline 6 \,\,\underline 6 \,\,\underline 6 \,\,\underline 6 \,\,\underline 6 \,\,\underline 6 \,\,\,\,\,\,\,,\,\,\,\,\underline {{m{viable}}} \,\,{m{solution}}\,\,\,\,\,\left( {{{666666} \over 7} = 95238} ight) \hfill \cr 
 n = 6\,\,\,\,\, \Rightarrow \,\,\,{m{impossible}}\,\,\,\,\,\left( {{m{idem}}\,\,n = 2} ight) \hfill \cr 
 n = 7\,\,\,\,\, \Rightarrow \,\,\,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\, \cdot \,\,\,\left( {7 + 2} ight) = \underline 8 \,\,\underline 8 \,\,\underline 8 \,\,\underline 8 \,\,\underline 8 \,\,\underline 8 \,\,\underline 8 \,\,\underline 8 \,\,\,\,\,,\,\,\,\,{m{impossible}}\,\,\,\,\,\left( {{{88888888} \over 9} 
e {\mathop{m int}} } ight) \hfill \cr 
 n = 8\,\,\,\,\, \Rightarrow \,\,\,{m{impossible}}\,\,\,\,\,\left( {{m{idem}}\,\,n = 2} ight) \hfill \cr 
 n = 9\,\,\,\,\, \Rightarrow \,\,\,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\underline {} \,\, \cdot \,\,\,\left( {9 + 2} ight) = \underline {10} \,\,\underline {10} \,\,\underline {10} \,\, \ldots \,\,\underline {10} \,\,\,????\,\,\,\,{m{impossible}}\, \hfill \cr 
 n \ge 10\,\,\,\,\, \Rightarrow \,\,\,\,{m{impossible}}\,\,\,\,\,\left( {{m{idem}}\,\,n = 9} ight) \hfill \cr} ight.

The correct answer is therefore (B). (This is all VERY fast, although hard to type!)

This solution follows the notations and rationale taught in the GMATH method.

Regards, 
Fabio.

ScottTargetTestPrep · Answer

Let’s test some values for n.

If n = 1, A is a 1-digit number. We multiply A by 3 and we get a 2-digit number, which is 22. However, 22 is not a multiple of 3. So n can’t be 1.

If n = 2, A is a 2-digit number. We multiply A by 4 and we get a 3-digit number, which is 333. However, 333 is not a multiple of 4. So n can’t be 2.

If n = 3, A is a 3-digit number. We multiply A by 5 and we get a 4-digit number, which is 4444. However, 4444 is not a multiple of 5. So n can’t be 3.

If n = 4, A is a 4-digit number. We multiply A by 6 and we get a 5-digit number, which is 55,555. However, 55,555 is not a multiple of 6. So n can’t be 4.

If n = 5, A is a 5-digit number. We multiply A by 7 and we get a 6-digit number, which is 666,666. We see that 666,666 is a multiple of 7 (666,666 = 7 x 95,238)! So n can be 5.

At this point, we can skip even values of n, since the (n+1)-digit number it forms is odd and will never be a number of n + 2, which is even.

If n = 7, A is a 7-digit number. We multiply A by 9 and we get a 6-digit number, which is 88,888,888. However, 88,888,888 is not a multiple of 9. So n can’t be 7.

If n = 9, A is a 9-digit number. We multiply A by 11 and we get a 10-digit number. However, we can’t have a 10-digit number in which each of its digits is 10. Therefore, n can’t be 9, and we can stop here.

There is only one instance, n = 5, where all the criteria are met.

Answer: B

EMPOWERgmatRichC · Answer

Hi Gmat_mission,

We're told that after multiplying a positive integer A (which has N digits) by (N+2), we get a number with (N+1) digits, all of whose DIGITS are (N+1). We're asked for the number of possible values of A that 'fit' this description.

To start, this question certainly 'feels' weird - and you would likely find it easiest to 'work back' from the later pieces of information that you're given - and use 'brute force' (along with some Number Properties) to find the solution. We're looking to end up with a number that has (N+1) digits - and ALL of those DIGITS equal (N+1). Since we're dealing with digits, there are only a limited number of possible values that we can end up with:

22 
333 
4444 
55555 
666666 
7777777 
88888888 
999999999

Thus, there are no more than 8 possibilities; to get the correct answer, we have to incorporate the other pieces of information that we're given and see which of these end numbers actually 'fits' everything that we're told. 
IF.... 
The end result was 22, then N=1, but there is no 1-digit number that you can multiply by (1+2) = 3 and end up with 22. This is NOT possible. 
The end result was 333, then N=2, but there is no 2-digit number that you can multiply by (2+2) = 4 and end up an ODD. This is NOT possible. 
The end result was 4444, then N=3, but there is no 3-digit number that you can multiply by (3+2) = 5 and end up with 4444. This is NOT possible. 
The end result was 55555, then N=4, but there is no 4-digit number that you can multiply by (4+2) = 6 and end up with ODD. This is NOT possible.

The end result was 666666, then N=5... there IS a 5-digit number that you can multiply by (5+2) = 7 and end up with 666666 (it's 95,238 - you just have to do a little division to prove it). This IS a possibility.

The end result was 7777777, then N=6, but there is no 6-digit number that you can multiply by (6+2) = 8 and end up with ODD. This is NOT possible. 
The end result was 88888888, then N=7, but there is no 7-digit number that you can multiply by (7+2) = 9 (since 88888888 is NOT a multiple of 9). This is NOT possible. 
The end result was 999999999, then N=8, but there is no 8-digit number that you can multiply by (8+2) = 10 and end up with ODD. This is NOT possible.

Thus, there's just one answer that 'fits' everything that we're told.

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

rvgmat12 · Answer

OE:

Correct answer: (B)

The numbers on the right hand side of the equation are restricted to 22, 333, 4444, 55555, 666666, 7777777, 88888888, and 999999999. We have a limited set of possible options. Looking at the example, we can try to find the 3 digit number, A, by dividing 4444 by 5. But since 4444 is not divisible by 5, there is no such positive integer.

Similarly, all we have to do is, using divisibility rules, divide each of these 9 numbers and see if it is divisible by its respective (n + 2). 22 is not divisible by 3; 333 is not divisible by 4; and so on. (Note: An odd number will never be divisible by an even number so we don’t need to check for odd numbers.)

We find only 666666 is divisible by 7 giving 95238 as A. So there is only one such A.

02999 · Answer

I got the way you solved the question. But how to determine if there is any other number that can be divided by (n+1). we will have to check manually for the first couple of numbers. is it?

Thanks!

bumpbot · Answer

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