What is the remainder when positive integer m is divided by 14?

Question

(1) When m is divided by 28, the remainder is 5
(2) When m is divided by 2, the remainder is 1 and when m is divided by 7, the remainder is 5

Source: OptimusPrep

I am on shaky grounds with remainder questions. For me, solving this type is always time consuming, so I was wondering whether there is an efficient method to solving this type of questions.

Bunuel · Answer

What is the remainder when positive integer m is divided by 14?

(1) When m is divided by 28, the remainder is 5 --> m is a multiple of 28 plus 5. Since 28 = 14*2, then m is a multiple of 14 plus 5, which means that m divided by 14 gives the remainder of 5. Sufficient.

(2) When m is divided by 2, the remainder is 1 and when m is divided by 7, the remainder is 5.

The first part (m is divided by 2, the remainder is 1) implies that m is odd.
The second part (m is divided by 7, the remainder is 5) means that m = 7q + 5 --> m can be 5, 12, 19, 26, 33, ... but since m must be odd, then m can only be 5, 19, 33, ... Each gives the remainder of 5 when divided by 14.

Sufficient.

Answer: D.

Hope it's clear.

Bunuel · Answer

Theory on remainders problems: remainders-144665.html
Tips on remainders: remainders-tips-and-hints-175000.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

naeln · Answer

Thank you Bunuel for the links and the clear explanation. Truly appreciate it :-D

MathRevolution · Answer

There is 1 variable (m) in the original condition. In order to match the number of variables and the number of equations, we need 1 equation. Since con 1) and con 2) each has 1 equation, there is a high chance that D is the correct answer. We can substitute the other parts. 
In case of con 1), if we divide m=28p+5=5, 33, 61,.... by 14, the remainder is always 5. The answer is unique and the condition is sufficient. 
In case of con 2), the number that satisfies m=2q+1=7n+5 is 5, which is the remainder of dividing m by 14. The answer is unique and the condition is sufficient. Hence, the correct answer is D.

- For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

bumpbot · Answer

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What is the remainder when positive integer m is divided by 14?

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What is the remainder when positive integer m is divided by 14?

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