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Math Expert V
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What is the tens digit of positive integer x ?  [#permalink]

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Question Stats: 75% (01:13) correct 25% (01:30) wrong based on 1886 sessions

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What is the tens digit of positive integer x ?

(1) x divided by 100 has a remainder of 30.
(2) x divided by 110 has a remainder of 30.

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Re: What is the tens digit of positive integer x ?  [#permalink]

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SOLUTION

What is the tens digit of positive integer x ?

(1) x divided by 100 has a remainder of 30 --> $$x=100q+30$$, so $$x$$ can be: 30, 130, 230, ... Each has the tens digit of 3. Sufficient.

(2) x divided by 110 has a remainder of 30 --> $$x=110p+30$$, so $$x$$ can be: 30, 250, ... We already have two values for the tens digit. Not sufficient.

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Re: What is the tens digit of positive integer x ?  [#permalink]

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St 1: Sufficient: X should be number such as 130, 230, 330, 430.... In all case if we divide X by 100 the remainder is 30. so the tens digit is 3
St 2: Insufficient: Let say X be 140 : In this case the remainder is 30 when divided by 110, So tens digit is 4
Let say X be 250: In this case the remainder is 30 when divided by 110, So tens digit is 5.

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The best way to solve this question is to use equation rather than getting stuck in remainder part
1) x = 100a+30 , where a is non-negative integer.
Thus x can take following values 30,130,230,330....etc
So the ten's digit always will be 3
Sufficient
2) x = 110a+30 , where a is non-negative integer.
Thus x can take following values 30,140,250,380....etc
So the ten's digit in not unique
In-Sufficient
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Re: What is the tens digit of positive integer x ?  [#permalink]

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What is the tens digit of positive integer x ?

(1) x divided by 100 has a remainder of 30.
Means, x=100n+30 ; n=1,2,3...
x can be 130,230,330-in all cases tens place is held by 3- sufficient

(2) x divided by 110 has a remainder of 30.
Means, x=110n+30 ; n =1,2,3...
x can be 110,220,330- the tens place is changing with changing value of n - insufficient

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Re: What is the tens digit of positive integer x ?  [#permalink]

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Bunuel wrote:
What is the tens digit of positive integer x ?

(1) x divided by 100 has a remainder of 30 --> x=100q+30: 30, 130, 230, ... as you can see every such number has 3 as the tens digit. Sufficient.

(2) x divided by 110 has a remainder of 30 --> x=110p+30: 30, 140, 250, 360, ... so, there are more than 1 value of the tens digit possible. Not sufficient.

Question 1) x divided by 100 has a remainder of 30 --> x=100q+30: 30, 130, 230, ... as you can see every such number has 3 as the tens digit. Sufficient. QUESTION :How exactly are 30, 130, 230 computed?
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sagnik242 wrote:
Bunuel wrote:
What is the tens digit of positive integer x ?

(1) x divided by 100 has a remainder of 30 --> x=100q+30: 30, 130, 230, ... as you can see every such number has 3 as the tens digit. Sufficient.

(2) x divided by 110 has a remainder of 30 --> x=110p+30: 30, 140, 250, 360, ... so, there are more than 1 value of the tens digit possible. Not sufficient.

Question 1) x divided by 100 has a remainder of 30 --> x=100q+30: 30, 130, 230, ... as you can see every such number has 3 as the tens digit. Sufficient. QUESTION :How exactly are 30, 130, 230 computed?

If q=0, then x=30;
If q=1, then x=130;
If q=2, then x=230;
If q=3, then x=330;
...

All these numbers when divided by 100 gives the remainder of 30.

Generally, if $$x$$ and $$y$$ are positive integers, there exist unique integers $$q$$ and $$r$$, called the quotient and remainder, respectively, such that $$y =divisor*quotient+remainder= xq + r$$ and $$0\leq{r}<x$$.

For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since $$15 = 6*2 + 3$$.

Notice that $$0\leq{r}<x$$ means that remainder is a non-negative integer and always less than divisor.

This formula can also be written as $$\frac{y}{x} = q + \frac{r}{x}$$.

You really need to brush up fundamentals:
Theory on remainders problems: remainders-144665.html

Units digits, exponents, remainders problems: new-units-digits-exponents-remainders-problems-168569.html

All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199

Hope this helps.
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Re: What is the tens digit of positive integer x ?  [#permalink]

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Bunuel chetan2u niks18
amanvermagmat

Quote:
What is the tens digit of positive integer x ?

(1) x divided by 100 has a remainder of 30 --> $$x=100q+30$$, so $$x$$ can be: 30, 130, 230, ... Each has the tens digit of 3. Sufficient.

(2) x divided by 110 has a remainder of 30 --> $$x=110p+30$$, so $$x$$ can be: 30, 250, ... We already have two values for the tens digit. Not sufficient.

Is it valid to take quotient(p/q) as 0 ? I assumed if divisor is divided by dividend, we need minimum value of quotient to be
non positive for division to be valid.
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Re: What is the tens digit of positive integer x ?  [#permalink]

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Bunuel chetan2u niks18
amanvermagmat

Quote:
What is the tens digit of positive integer x ?

(1) x divided by 100 has a remainder of 30 --> $$x=100q+30$$, so $$x$$ can be: 30, 130, 230, ... Each has the tens digit of 3. Sufficient.

(2) x divided by 110 has a remainder of 30 --> $$x=110p+30$$, so $$x$$ can be: 30, 250, ... We already have two values for the tens digit. Not sufficient.

Is it valid to take quotient(p/q) as 0 ? I assumed if divisor is divided by dividend, we need minimum value of quotient to be
non positive for division to be valid.

What is the remainder when 30 is divided by 100? Isn't it 30? So, why could not p (or q) be 0?
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Re: What is the tens digit of positive integer x ?  [#permalink]

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Bunuel wrote:
What is the tens digit of positive integer x ?

(1) x divided by 100 has a remainder of 30
(2) x divided by 110 has a remainder of 30

Target question: What is the tens digit of positive integer x ?

Statement 1: x divided by 100 has a remainder of 30
ASIDE: When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.
In other words, the possible values of k are: 1, 6, 11, 16, 21, 26, 31, . . . etc.

So, from statement 1, we can conclude that the possible values of x are: 30, 130, 230, 330, 430, 530,. . .
In ALL possible values of x, the tens digit is always 3
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x divided by 110 has a remainder of 30
From statement 2, we can conclude that the possible values of x are: 30, 140, 250, 360, 470, 580,. . .
Notice that the the tens digit can have many different values
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Cheers,
Brent
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_________________ Re: What is the tens digit of positive integer x ?   [#permalink] 07 Apr 2020, 10:01

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