What is the remainder when n is divided by 15?

Question

(1) When n is divided by 5 the remainder is 4
(2) When n is divided by 6 the remainder is 5.

Bunuel · Answer

GMAT wording would be: If n is a positive integer , what is the remainder when n is divided by 15? (1) When n is divided by 5 the remainder is 4 --> n=5q+4 , so n can be: 4, 9, 14, 19, 24, 29 , ... Not sufficient to get the single numerical value of the remainder. (2) When n is divided by 6 the remainder is 5 --> n=6p+5 , so n can be: 5, 11, 17, 23, 29 , ... Not sufficient to get the single numerical value of the remainder. (1)+(2) General formula of n based on the above two conditions would be n=30t+29 , divisor will be the least common multiple of above two divisors 5 and 6, hence 30 and the remainder will be the first common integer in above two patterns, hence 29 (for more on this check: manhattan-remainder-problem-93752.html?hilit=derive#p721341 and when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html?hilit=derive#p722552). So, we have that n=30t+29 --> n=30t+29=30t+15+14=15(2t+1)+14 , hence the remainder upon division of n by 15 is 14. Sufficient. Answer: C. Similar problems about the same concept: positive-integer-n-leaves-a-remainder-of-4-after-division-by-93752.html if-n-is-a-positive-integer-greater-than-16-is-n-a-prime-129829.html when-positive-integer-x-is-divided-by-5-the-remainder-is-128470.html when-n-is-divided-by-5-the-remainder-is-2-when-n-is-divided-82624.html when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html when-the-positive-integer-a-is-divided-by-5-and-125591.html what-is-the-value-of-length-n-100-meter-of-wire-126500.html a-group-of-n-students-can-be-divided-into-equal-groups-of-126384.html when-the-positive-integer-a-is-divided-by-5-and-7-the-104480.html positive-integer-n-leaves-a-remainder-of-4-after-division-by-93752.html when-positive-integer-n-is-divided-by-3-the-remainder-is-86155.html Hope it helps.

mamba5 · Answer

Hi Bunuel,

For these types of questions...how can a possibly integer for X be the remainder itself?

I.e. From Manhattan's book they say "When positive integer X is divided by 5, the remainder is 2"

The possibilities they list for X are 2, 7, 12, 17.

I don't see how 2 is a possibility.

I am seeing this happen on similar questions so I think I'm missing a fundamental rule or something.

Bunuel · Answer

THEORY:
 Positive integer  a  divided by positive integer  d  yields a reminder of  r  can always be expressed as  a=qd+r , where  q  is called a quotient and  r  is called a remainder, note here that  0\leq{r}<d  (remainder is non-negative integer and always less than divisor).

For example positive integer n is divided by 25 yields the remainder of 13 can be expressed as:  n=25q+13 . Now,  the lowest value of  q  can be zero  and in this case  n=13  --> 13 divided by 25 yields the remainder of 13.  Generally when divisor (25 in our case) is more than dividend (13 in our case) then the reminder equals to the dividend . For example:
3 divided by 24 yields a reminder of 3 -->  3=0*24+3 ;
or:
5 divided by 6 yields a reminder of 5 -->  5=0*6+5 ,
or example from your post:
2 divided by 5 yields a reminder of 2 -->  2=0*5+2 .

Hope it's clear.

MariaS · Answer

Bunuel, Is the following solution valid?

What is the remainder when n is divided by 15?

(1) When n is divided by 5 the remainder is 4 
(2) When n is divided by 6 the remainder is 5.

Clearly, both statements are insufficient separately, but taken together, we know that when n is divided by 30 (5*6) the remainder is 20 (4*5) --> in other words, when n is divided by 15. the remainder is 10.

Bunuel · Answer

No, that's not correct. If you refer to the solution above you'll see that when we combine two statements formula for n becomes n=30t+29 , so when n is divided by 30 the remainder is 29, not 20 ( n could be 29, 59, 89, 119, ...). From there: n=30t+29=30t+15+14=15(2t+1)+14 , hence the remainder upon division of n by 15 is 14, not 10. Complete solution here: what-is-the-remainder-when-n-is-divided-by-131442.html#p1079249 Also please follow the links provided in that post to see similar questions with explanations. Finally, even if we had that when n is divided by 30 the remainder is 20 ( n=30q+20 ) still your conclusion would be wrong: n=30q+20=30q+15+5=15(2q+1)+5 , hence in this case the remainder upon division of n by 15 would be 5, not 10. Hope it's clear.

What is the remainder when n is divided by 15?

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What is the remainder when n is divided by 15?

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