How many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divide

BTGModeratorVI wrote: ↑

Wed Feb 03, 2021 10:36 am

How many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14?

A. 3
B. 4
C. 5
D. 6
E. 7

Answer: A
Source: Veritas Prep

When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Two-digit number yields a remainder of 1 when divided by 14.
So, the possible values are: 15, 29, 43, 57, 71, 85 and 99
At this point, we have 7 possible values

Two-digit number yields a remainder of 1 when divided by 4.
Examine each of the 7 values and determine which ones yield a remainder of 1 when divided by 4
They are: 15, 29, 43, 57, 71, 85 and 99

So, there are 3 values that satisfy BOTH conditions.

Answer: A

BTGModeratorVI wrote: ↑

Wed Feb 03, 2021 10:36 am

How many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14?

A. 3
B. 4
C. 5
D. 6
E. 7

Answer: A

Solution:

The smallest positive integer that is divisible by both 4 and 14 is their LCM, which is 28. Thus 28 + 1 = 29 will yield a remainder of 1 when divided by both 4 and 14. We can add 28 to each previous dividend to obtain additional dividends:

29 + 28 = 57

57 + 28 = 85

Since the next number will be 85 + 28 = 113, which is a 3-digit number, we can stop at 85. Thus, we have three 2-digit numbers (29, 57, and 85) that will yield a remainder of 1 when divided by both 4 and 14.

Answer: A