any shorter method to solve this

[anjaligeorge1]

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3 ?

A. 15
B. 16
C. 17
D. 18
E. 19

[Stuart@KaplanGMAT]

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3 ?

A. 15
B. 16
C. 17
D. 18
E. 19

There are 51 integers in the range. Since 51 is divisible by 3, the answer will be 17 (there will be an equal number of integers with remainder 0, 1 and 2).

If the number of integers hadn't been divisible by 3, then we'd have to be more careful.

For example, if the range had been 1 to 50, then the answer would be either 16 or 17 and to figure it out completely we'd look at the first 2 integers and see if one of them gave the remainder we wanted (we'd only look at the first 2, because the remaining 48 can be grouped into 16 sets of 3, so they'll be balanced).

Since 1 gives a remainder of 1 when we divide by 3, the answer for the inclusive set 1 to 50 would be 1 + 48/3 = 1 + 16 = 17.

[smashinonions]

well, the range of numbers is from 0 - 50 so we have (50 -0) + 1 = 51 numbers but the last number that has a remainder of 1 when divided by 3 is 49 ( 49 / 3 = 16 with remainder 1)

now, here the numbers from 0 - 50 which leave a remainder of 1 when divided by 3 can be written as

1, 4, 7, 10, 13...

here d = 3
last term is 49

so 49 = 1 + (n - 1) * 3

(49 - 1)/ 3 = n - 1

48/ 3 = n - 1

16 = n - 1

n = 16 + 1

n = 17



i.e we have 17 numbers

[erjamit]

But how come 1 divided by 3 will give a remainder of 1...i am bit confused by this.

[sudhir3127]

A number can be written as = divisor* quotient + remainder

hence 1 when divided by 3 can be written as 3*0+ 1

hope its clear now...

[erjamit]

I understand that....but then how do we differentiate the modulus operation from division....

[Ian Stewart]

I understand that....but then how do we differentiate the modulus operation from division....

That depends- what do you mean by 'modulus'? The word modulus is used in several different ways in mathematics: it can refer to absolute value (which has very little to do with division; absolute value measures distances on the number line), or it can refer to congruences in modular arithmetic (which are entirely based on properties of division), among other things.

If you are thinking of modular arithmetic (which can be helpful on the GMAT, but which you certainly don't need to know), when we say that, for example "7 is congruent to 1, mod 3" we mean "7 has the same remainder as 1 when we divide by 3". Numbers are congruent precisely because they have the same remainder, so modular arithmetic is fundamentally based on division. If anyone reading this doesn't know what I'm talking about, don't worry about it- it's not needed for the GMAT.

[erjamit]

Ian,

1 mod 3 is 1. (Modular arthimetic).

if 1 divided by 3 gives a remainder of 1 then what is the difference between mod and division. They are both the same then. But in computer science/mathematics they are different.

I hope you are getting my point here.

Amit

[Ian Stewart]

Ian,

1 mod 3 is 1. (Modular arthimetic).

if 1 divided by 3 gives a remainder of 1 then what is the difference between mod and division. They are both the same then. But in computer science/mathematics they are different.

I hope you are getting my point here.

Amit

Modular arithmetic is simply arithmetic with remainders. It is different, I suppose, from division, because when we're doing modular arithmetic, we don't care at all about quotients, only remainders. When we say "1 mod 3 is 1", that simply means "the remainder is 1 when 1 is divided by 3". When we say that "26 = 20 mod 3", that just means "26 and 20 have the same remainder when divided by 3". This can be taught in different ways- you may have learned it differently, particularly if you learned it in Computing Science- but modular arithmetic is fundamentally based on remainders. Perhaps you could explain what you think the differences are?

[erjamit]

Hi Ian,

It is getting bit confusing right now. But I don't want to spend more time on it (just 2 weeks left).

So for GMAT what I have learnt is

1/3 will leave a remainder of 1, 2/3 will leave 2 etc.

If that is the case let it be. I will just rote it.

Thanks
Amit

[Stuart@KaplanGMAT]

Hi Ian,

It is getting bit confusing right now. But I don't want to spend more time on it (just 2 weeks left).

So for GMAT what I have learnt is

1/3 will leave a remainder of 1, 2/3 will leave 2 etc.

If that is the case let it be. I will just rote it.

Thanks
Amit

That's correct.

In fact, another way you can think of a remainder is as the numerator in a (non-reduced) mixed fraction. (Note: this only works for positive numbers, which is pretty much all you'll see on GMAT remainder questions.)

20/3 = 6 and 2/3.. so 20 divided by 3 has a remainder of 2

40/7 = 5 and 5/7... so 40 divided by 7 has a remainder of 5

3/7 = 0 and 3/7... so 3 divided by 7 has a remainder of 3

[erjamit]

WOW...thats grt...explanation.....

Thanks Stuart, however I would like to discuss more on this after my GMAT exam.

Amit

[shashank.ism]

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3 ?

A. 15
B. 16
C. 17
D. 18
E. 19

shortest method possible is
if we arrange this in AP, we get 1+4+7+10+.......+49 so 1+(n-1)3=49: n=17 Ans C

Well for the case of how you can get 1 as remainder when 1 is divisble by 3. Simple funda of remainder is whatever left after dividing a number. here you try to divide one by 3. since it is not divisible quotient=0 and remainder =1

[mgmt_gmat]

50> 3k+1

from 1 to 16 multiple of 3.. hence 16 + 1(for number 1 when divided by 3 leaves remainder of 1)

[komal]

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3 ?

A. 15
B. 16
C. 17
D. 18
E. 19

its basically 3K+1
hence series is 1,4,7,........,49
hence n=17
C. 17