Problem Solving

Vincen wrote:For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT:

A. 32
B. 38
C. 44
D. 52
E. 63

Great question!!

USEFUL PROPERTY:
When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0
Conversely, if I know that, when k is divided by w, the remainder is 5, then I know that w must be greater than 5

The remainder when q is divided by r is 7
This tells us that r is greater than 7

s is divided by t is 3
This tells us that t is greater than 3

Now check the answer choices...

A) 32
Is it POSSIBLE for rt to equal 32?
Yes, if r = 8 and t = 4, then rt = 32
ELIMINATE A

B) 38
Is it POSSIBLE for rt to equal 38?
NO.
There are only two ways to write 38 as the product of POSITIVE INTEGERS:
i) (2)(19) = 38
ii) (1)(38) = 38
If r is greater than 7 and t is greater than 3, there's no way that one of the values (r or t) can equal 1 or 2.

Answer: B

Cheers,
Brent

Vincen wrote:For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT:

A. 32
B. 38
C. 44
D. 52
E. 63

OA is B.

To solve this problem, we need a clear insight on division between integers. Recall that in any division, the divisor must be greater than the remainder. Thus, we know that r > 7 and t > 3. In the other words, r ≥ 8 and t ≥ 4. Thus, rt is at least 8(4) = 32. Looking at the answer choices, all of them are at least 32. Therefore, we need further analysis. One thing we can do is to factor each answer choice:

32 = 1(32) = 2(16) = 4(8)

38 = 1(38) = 2(19)

44 = 1(44) = 2(22) = 4(11)

52 = 1(52) = 4(13)

63 = 1(63) = 3(21) = 7(9)

We see that all the answer choices except 38 can be expressed as a product of two factors where one factor is greater than 7 and the other is greater than 3. For example, rt could be 44 since r could be 11 and t could be 4. The only answer choice that doesn't have this property is 38. Thus, 38 is the correct answer.

Answer: B