A, B and C are different prime numbers, and positive integer D is a no

Question

A, B and C are different prime numbers, and positive integer D is a non-prime number. If D divided by C equals B with remainder A, what is the smallest possible value of B + C?

A) 8
B) 10
C) 12
D) 14
E) 18

chetan2u · Answer

You can have umpteen values for A, B , C and D..
Now,  D=B*C+A 
One restriction is that A has to be smaller than C, so let it be 2, then B and C can be 3 and 5.
Thus  D=3*5+2=17 , but 17 is prime so D turns out to be prime.
Similarly, when B and C are 3 and 7,  D=3*7+2=23 , but 23 is prime so D turns out to be prime.

Next possibility is when A=3, and B and C can be checked for other values. 
 But our answers are even, so B and C both must be even.. 
 Also in every case BC+A or D will be EVEN when A, B and C are ODD.
Hence  B and C can take next larger values after A or 3, so B and C are 5 and 7 , and 5+7=12..

But say when A=5, and B and C are 3 and 7, so D=3*7+5=26...possible

B

Another way to avoid mistakes( as I missed out a case above) and a short method would be 
 use choices.. 
A) 8
So B=3 and C=5..A can be only 2, so D=3*5+2=17, a prime number..wrong
B) 10
So B=3 and C=7, so A can be 2 or 5...
A=2, D=3*7+2=23..No 
A=5, D=3*7+5=26..YES

We get our answer as 10

firas92 · Answer

chetan2u

How about A=5, B=3, C=7, D=26? A just has to be smaller than C

26=3*7+5

In which case B+C=10

chetan2u · Answer

HI
you are correct. I did check for A as 3 and B as 2 with C as 5 and 7, but missed out checking for A=5.
Kudos

Camach700 · Answer

We know that

D= B * C + A
A<C

take B=3; C=5 and A=3
--> 18 = 3 * 5 + 3 --> min(B +C) = 8

All restrictions are satisfied, is there anything wrong in my reasoning?

BrentGMATPrepNow · Answer

Good idea!
The problem is that the question says "A, B and C are  different  prime numbers"

ashdank94 · Answer

Why cant A be 7 when I take B as 3 and C as 5? 
15 + 7 gives me 22 which is not prime

Posted from my mobile device

chetan2u · Answer

A has to be less than C.

D divide by C or 5 cannot give A or 7 as remainder. Remainder has to be less than the divisor or 5 here.

Kinshook · Answer

Given: A, B and C are different prime numbers, and positive integer D is a non-prime number.
Asked: If D divided by C equals B with remainder A, what is the smallest possible value of B + C?

D = BC + A

Prime numbers = {2,3,5,7,11,13,17,19,23,29,}

For the smallest possible value of B+C, BC = 2*3 = 6; B+C=5; No value of A is possible; Rejected
For next smallest possible value of B+C, BC = 2*5 = 10; B+C=7; A=3; D= 13; Rejected
For next smallest possible value of B+C, BC = 3*5 = 15; B+C=8; A=2; D=17; Prime; Rejected
For next smallest possible value of B+C, BC = 2*7 = 14; B+C=9; A=3; D=17; Prime; Rejected; A=5;D=19;Prime; Rejected
For next smallest possible value of B+C, BC = 3*7 = 21; B+C=10; A=2; D=23; Prime; Rejected; A=5;D=26=2*13; Non-prime: Accepted

26 = 3*7 + 5
D = 26; C=7; B=3; A=5; B+C = 3+7 = 10

​​​​​​​IMO B

bumpbot · Answer

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A, B and C are different prime numbers, and positive integer D is a no

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A, B and C are different prime numbers, and positive integer D is a no

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