When 34, 36, and 38 are each divided by positive integer d, the remainders are 4, 0, and 2, respectively. When 41, 43, a

BTGModeratorVI wrote: ↑

Wed Jan 06, 2021 8:15 am

When 34, 36, and 38 are each divided by positive integer d, the remainders are 4, 0, and 2, respectively. When 41, 43, and 45 are each divided by positive integer k, the remainders are 5, 7, and 0, respectively. What is the value of d + k?

A) 12
B) 15
C) 18
D) 21
E) 24

Answer: B
Source: official guide

----ASIDE----------------
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
-----------------------------
If 34 divided by d leaves a reminder of 4, then (according to the above property), d > 4
If 36 divided by d leaves a reminder of 0, then d is a divisor of 36.
The divisors of 36 are: 1, 2, 3, 4, 6, 9, 12, 24 and 36
Since we already know that d > 4, d must be 6, 9, 12, 24 or 36. Let's test these five possible d-values

If d = 6, then 34, 36, and 38 divided by d leaves reminders 4, 0, and 2, respectively. PERFECT!!
At this point, we can stop examining other possible values of d (since the answer choices tell us there can be only one correct answer)
So d = 6


If 43 divided by k leaves a reminder of 7, then (according to the above property), k > 7
If 45 divided by k leaves a reminder of 0, then k is a divisor of 45.
The divisors of 45 are: 1, 3, 5, 9, 15 and 45
Since we already know that k > 7, k must be 9, 15 or 45. Let's test these three possible k-values
If k = 9, then 41, 43, and 45 divided by d leaves reminders 5, 7, and 0, respectively. PERFECT!!
So k = 9

What is the value of d + k?
d + k = 6 + 9 = 15

Answer: B

BTGModeratorVI wrote: ↑

Wed Jan 06, 2021 8:15 am

When 34, 36, and 38 are each divided by positive integer d, the remainders are 4, 0, and 2, respectively. When 41, 43, and 45 are each divided by positive integer k, the remainders are 5, 7, and 0, respectively. What is the value of d + k?

A) 12
B) 15
C) 18
D) 21
E) 24

Answer: B
Source: official guide

Solution:

We see that d = 6 since 34/6 = 5 R 4, 36/6 = 6 R 0 and 38/6 = 6 R = 2. We see that k = 9 since 41/9 = 4 R 5, 43/9 = 4 R 7 and 45/9 = 5 R 0. Therefore, the value of d + k = 6 + 9 = 15.

(Note: One can come up with the value of d being 6 since d has to be a divisor of 36 and greater than 4 (recall that divisor is greater than remainder). This limits the value of d to be 6, 9, 12, 18, or 36. By simply dividing by 6, we see that d must be 6. Likewise, one can come up with the value of k being 9 since k has to be a divisor of 45 and greater than 7. This limits the value of k to be 9, 15, or 45. By simply dividing by 9, we see that k must be 9.)

Answer: B