The smallest number which, when divided by 4, 6 or 7 leaves a remainde

Question

The smallest number which, when divided by 4, 6 or 7 leaves a remainder of 2, is:

a) 44
b) 62
c) 80
d) 86
e) none of these

medanova · Answer

I first find the smallest possible number which is divisible by 4,6 and 7. With a prime factorization you can easily find that this number is 84.

Therefore, 86 (84+2) will be the smallest possible number which when divided by 4, 6 or 7 leaves a remainder of 2. Hence answer should be D.

siyer · Answer

The smallest number which, when divided by 4, 6 or 7 leaves a remainder of 2, is:
a) 44
b) 62
c) 80
d) 86
e) none of these

A number will give a remainder only when it is greater than the multiple of the divisors.
In such problems, always start with the greatest divisor. In our case 7.

Test each number with 7. We are left with 44 & 86.
Now test 44 & 86 with 6. We are left with 44 & 86.
Now test 44 & 86 with 4. We are left with 86.

Alternatively after testing with 7 we can easily see that 44 is a multiple of 4. So we only test 86 with the remaining options.

Hope this helps.

shrouded1 · Answer

Right, the solutions above are correct in their own way. But the way I think about this is the following :

The set of numbers that are divisible by numbers a,b,c are {LCM,LCMx2,LCMx3,...}
So what we need to know is the LCM of 4,6,7
LCM of these numbers is 84.

So the smallest number leaving remainder 2 will be 86

Answer : (d)

vitaliyGMAT · Answer

I have another vision of this problem:

In order to leave same remainder upon division by 4, 6 or 7 our number should be in the form:

N = LCM (4, 6, 7)*x + 2 , where  x >= 0

N = 84*x + 2

And we have AP starting from 2

2, 86, 170 ...

Smallest number in progression which leaves remainder 2 when divided by all of the 4, 6 and 7 is 2 not 86.

Hence answer should be E.

Bunuel please correct me if I'm wrong, may be I missed something.

KarishmaB · Answer

Note that the smallest such number is 2. 
When 2 is divided by 4, the quotient is 0 and remainder is 2. 
Same logic for 6 and 7 too.

Out of the given options, the next smallest number will be LCM (4, 6, 7) + 2 = 86

Answer (D)

vitaliyGMAT · Answer

Dear Karishma

Thank you very much for your reply.

But why quotient should not be 0. The question asks about remainder not specifying the quotient. Why the whole number can not go into remainder without splitting into quotient and remainder? I can’t still get why 2 is not appropriate? Question asks about min number.

Could you please elaborate.

Thank you in advance

vitaliyGMAT · Answer

Dear Karishma

Little more

The Number is in the form:

N=LCM(a,b,c)*n + 2

Why can't we take n=0?

vitaliyGMAT · Answer

let-s-be-the-set-of-all-positive-integers-that-when-divided-by-8-hav-187551.html

In the question above in order to get oficial answer A we need to strat our progression from remainder itself (5) and our quotient is 0.

Can anybody explain why we can't use 2 as our answer here?

I think there should be phrazes like "different from remainder itself", "two digit number" or A<N<B in order to indicate some interval.

Many thanks and kudos for explanation from me.

prashantrchawla · Answer

Another way of looking at the problem, 
Going directly to the answer options, two answers can be omitted directly as they are divisible by 4 i.e. 44 and 80. We want the answer where after division from 4 remainder should be 2. 
For remaining two should comply the equation 4x+2, 6x+2 and 7x+2.
The answer is 86. Answer D.

vitaliyGMAT · Answer

Dear Prashantrchawla Your approach is good and 86 definitely leaves remainder 2 when divided by 4, 6 and 7. But, I have some doubts about this question. I think it’s trickier than it seems. The question does not ask which number from the list leaves remainder … or which smallest two digit number leaves remainder … or something of that sort. It simply and directly asks “what smallest number …” and smallest number is definitely 2 not 86. But official answer says 86 so let it be. This will remain my personal dilemma

. Thanks for kudos by the way

KarishmaB · Answer

The answer should be 2. The quotient can be 0. I said in my post above that the smallest such number is 2. The options don't have 2 and hence we needed to ignore it. Else, the correct answer is 2 only.

vitaliyGMAT · Answer

Dear Karishma
Thanks a lot for confirmation

MvArrow · Answer

IMO the correct answer should be E as the smallest number is 2!

bumpbot · Answer

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The smallest number which, when divided by 4, 6 or 7 leaves a remainde

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