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How many positive two-digit numbers yield a remainder of 1 when divide

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How many positive two-digit numbers yield a remainder of 1 when divide [#permalink]

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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
[Reveal] Spoiler: OA

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Re: How many positive two-digit numbers yield a remainder of 1 when divide [#permalink]

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New post 07 Mar 2017, 11:30
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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


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 = 29
29 + 14 = 43
43 + 14 = 57
57 + 14 = 71
71 + 14 = 85
85 + 14 = 99

The answer is (A) 3.
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Re: How many positive two-digit numbers yield a remainder of 1 when divide [#permalink]

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New post 07 Mar 2017, 11:34
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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Re: How many positive two-digit numbers yield a remainder of 1 when divide [#permalink]

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New post 08 Mar 2017, 00:18
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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,
an alternative solution would be:

the LCM of 4 & 14 is 28 therefore the numbers are 28+1, 56+1, 84+1.

thanks.
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Re: How many positive two-digit numbers yield a remainder of 1 when divide [#permalink]

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First we need to find the numbers which are divisible by 4 and 14. the numbers are oing to be LCM of the 4 and 14 which will be 28, 56, 84.

hence the number leaving 1 as remanider will be 28+1, 56+1 and 84+1
Option A
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Re: How many positive two-digit numbers yield a remainder of 1 when divide [#permalink]

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New post 08 Mar 2017, 01:51
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First we need to find the numbers which are divisible by 4 and 14. the numbers are oing to be LCM of the 4 and 14 which will be 28, 56, 84.

hence the number leaving 1 as remainder will be 28+1, 56+1 and 84+1
Option A
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Re: How many positive two-digit numbers yield a remainder of 1 when divide [#permalink]

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New post 07 Apr 2018, 20:48
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


least value of n=1
next least value=29
1+(29-1)x<100
x<3.5
x=3
A
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Re: How many positive two-digit numbers yield a remainder of 1 when divide [#permalink]

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New post 08 Apr 2018, 06:38
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Hi,
an alternative solution would be:

the LCM of 4 & 14 is 28 therefore the numbers are 28+1, 56+1, 84+1.

thanks.
Re: How many positive two-digit numbers yield a remainder of 1 when divide   [#permalink] 08 Apr 2018, 06:38
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How many positive two-digit numbers yield a remainder of 1 when divide

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