Bunuel wrote:

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

I'd start with the ones that have a remainder of 1 when divided by 14. There'll be fewer of those, so it'll be much quicker to list them out.

14 + 1 = 15

15 + 14 = 29

29 + 14 = 43

43 + 14 = 57

57 + 14 = 71

71 + 14 = 85

85 + 14 = 99

Notice that I'm not taking the slow route of doing '14*2+1, 14*3+1, 14*4+1...' etc. I find that tends to be slower unless you happen to have the multiples of 14 memorized, since multiplication is a bit slower than addition. A nice shortcut for listing numbers with a particular remainder, is to start with the remainder (1), then count up by the divisor (in this case 14) repeatedly.

Okay, we have our list. How many of those have a remainder of 1 when divided by 4? The first thing I notice is that 15 has a remainder of 3 when divided by 4, but 29 has a remainder of 1. Why? Well, it looks like when the 14 is multiplied by 2, it becomes a multiple of 4 (28). So, adding 1 to it will give you a number that also has a remainder of 1 when you divide it by 4. On the other hand, if the 14 isn't multiplied by 2, it isn't a multiple of 4. So, adding 1 to it won't give you a 'good' number.

That means every other value on our list will work.

14 + 1 = 15

15 + 14 =

2929 + 14 = 43

43 + 14 =

5757 + 14 = 71

71 + 14 =

8585 + 14 = 99

The answer is (A) 3.

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