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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.

Re: How many positive two-digit numbers yield a remainder of 1 when divide
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07 Mar 2017, 10:34

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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

Hi,

Take the numbers which are multiple of 14-28,42,56,70,84,98,112,126--Out of which only 28,56,84 are divisible by both 14 & 4 hence if you add 1 to these numbers. It will give you remainder of 1 when divided by 4 and remainder of 1 when divided by 14. Hence 3
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12 Sep 2018, 17:41

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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

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.

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17 Sep 2018, 17:26

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

We can use the fact that if a number n leaves a remainder of r when divided by a and b (a ≠ b), then n leaves a remainder of r when divided by the least common multiple (LCM) of a and b.

Since the LCM of 4 and 14 is 28, the smallest such two-digit number is 29 (notice that 29/4 = 7 R 1 and 29/14 = 2 R 1). The next two-digit number is 29 + LCM = 29 + 28 = 57. The next one is 57 + 28 = 85. Since the next one will be more than 100, we can stop at 85. Therefore, we see that there are 3 such integers.

Answer: A
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