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# Can anyone explain to me ?

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Can anyone explain to me ? [#permalink]  18 Jun 2010, 07:38
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If S and T are positive integers such that S/t=64.12, which of the follwing could be the remainder when s is divided by T ?

A 2
B 4
c 8
D 20
E 45

the answer is E, But why ?

Of the three-digit integers greater than 700, how many have two digits that are equal to each other and the remaining different from the other two ?

A 90
B 82
C 80
D 45
E 36

The answer is C, again I dont know why ? so pleas someone help me out and I will very thankful
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Re: Can anyone explain to me ? [#permalink]  18 Jun 2010, 07:53
Expert's post
If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t?
(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

We know that when we are told that "s divided by t gives remainder r" can be expressed by the following formula: $$s=qt+r$$, in or case $$q=64$$, --> $$s=64t+r$$, divide both parts by $$t$$ --> $$\frac{s}{t}=64.12=64+\frac{r}{t}$$ --> $$0.12=\frac{r}{t}$$--> $$\frac{3}{25}=\frac{r}{t}$$ so $$r$$ must be the multiple of 3. Only answer multiple of 3 is 45.

Or: $$\frac{s}{t}=64\frac{12}{100}=64\frac{3}{25}$$, so if the divisor=t=25 then the remainder=r=3. Basically we get that divisor is a multiple of 25 and the remainder is a multiple of 3. Only answer multiple of 3 is 45.

Of the three digits greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two
(A) 90
(B) 82
(C) 80
(D) 45
(E) 36

3-digit numbers more than 700 = 299;

Numbers with all distinct digits = 3*9*8=216 (first digit can take 3 values - 7, 8, 9; second digit can take 9 values and the third digit 8 values);

Numbers with all the same digits more than 700 = 3, (777, 888, 999);

{all} - {all distinct} - {all same} = {two equal} --> 299-216-3=80.

Please post one question per topic.
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Re: Can anyone explain to me ? [#permalink]  18 Jun 2010, 10:26
^
For Q1. I had 3/25T=R... but I couldn't figure out what to do from here...

It maybe easier to rewrite it as T=R*25/3. Since R must be an integer, it must be divisible by 3.
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Re: Can anyone explain to me ? [#permalink]  30 Jun 2010, 16:30
I need someone to explain the questions with datialed but thanks for the help anyway
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Re: Can anyone explain to me ? [#permalink]  01 Jul 2010, 12:11
xmagedo wrote:
I need someone to explain the questions with datialed but thanks for the help anyway

you know S/T= 64 + .12

.12= R/T R/T=12/100 -> R/T=3/25 -> 25R = 3T , because 25 does not have any factors of 3 in it that means that R must have a factor of 3 in it because 25 x R = a multiple of 3. Try plugging in real numbers if this seems fuzzy.

The second one is a combinators problem. The only way you can get the answer is by counting the possibilties.

999-700 = 299 total numbers

at this point you can find the number of ways where two equal digit occurs for 3 digit numbers or you can find number of ways that two equal digits do not occur subtract it from the total number of possibilitys

There are 3 possibilities for the first number 7,8,9
There are 10 possibilities for the second 0-9
There are 10 possibilities for the third 0-9

A 3 digit number can be 1 of 3 things in this case. It can have no equal digits, It can have 1 equal digit or it can have 2 equal digits. For example a number can be 707, 770, or 777. The possibility of having only 2 equal digit is as Bunuel mentioned "

{all} - {all distinct} - {all same} = {two equal}}

All = 299
All Distinct = 3x9x8 ( 3 for the first choice, the pool of choices is reduced by 1 for the second number, the pool is reduced again for the third - because we want DISTINCT numbers.

All same = 3 There only 3 numbers greater than 700 that are all the same here: they are 777,888, and 999

so - > 299-216-3=80.

You can solve this problem another way by finding directly how many numbers have 2 equal digits but its much harder.
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Re: Can anyone explain to me ? [#permalink]  03 Jul 2010, 11:33
xmagedo wrote:
I need someone to explain the questions with datialed but thanks for the help anyway

Could you be more specific as to what you don't understand? Which question? Where did Bunuel's answer lose you?
Re: Can anyone explain to me ?   [#permalink] 03 Jul 2010, 11:33
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