The remainder is 8 when 200 is divided by a positive integer 'x'.

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Quant Warrior

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The remainder is 8 when 200 is divided by a positive integer 'x'. How many possible values of "x" are there ?

My efforts to solve this question:

I broke the number like this: 192 + 8. One thing that I concluded instantly is that 'x' will be larger than 8 since 8 is remainder. Now I have to find out all the factors of 192. The number of factors of 192 is 14. From 14, I subtracted common factors of 200 and 192. The number of common factors is 4(1,2,4,8). So the number of values that can divide 200 leaving remainder 8 is: 14-4=10.

I am not sure if I am conceptually correct.

The answer in the book is 8.

Please help me with concept.

Regards.
 
Quant Warrior said:
The remainder is 8 when 200 is divided by a positive integer 'x'. How many possible values of "x" are there ?

My efforts to solve this question:

I broke the number like this: 192 + 8. One thing that I concluded instantly is that 'x' will be larger than 8 since 8 is remainder. Now I have to find out all the factors of 192. The number of factors of 192 is 14. From 14, I subtracted common factors of 200 and 192. The number of common factors is 4(1,2,4,8). So the number of values that can divide 200 leaving remainder 8 is: 14-4=10.

I am not sure if I am conceptually correct.

The answer in the book is 8.

Please help me with concept.

Regards.
Click to expand...
What about the larger than 8 requirement?
 
Saying "the remainder is 8 when 200 is divided by x" is the same as saying that 200= nx+ 8 for some integer "n". So nx= 200- 8= 192. Now, look at the ways of factoring 192 into two factors.

192= 2(96)= 2(2)(48)= 2(2)(2)(24)=2(2)(2)(2)(12)= 2(2)(2)(2)(2)(6)= 2(2)(2)(2)(2)(2)(3)= 2^6(3) is the "prime factorization" of 192.
One factorization into 2 factors is 64(3).
The others are just moving factors of 2 to the multiply the 3:
32(6)
16(12)
8(24)
4(48)
2(96)

So there are 12 different values for "x": 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, and 96 that are factors of 192. However, the fact that the remainder when 200 is divided by x is 8 requires that x be larger than 8. That reduces to 7 possible values for "x": 12, 16, 24, 32, 48, 64, and 96.

200/12= 16 with remainder 8
200/16= 12 with remainder 8
200/24= 8 with remainder 8
200/32= 6 with remainder 8
200/48= 4 with remainder 8
200/64= 3 with remainder 8
200/96= 2 with remainder 8

Cubist has reminded me that "192" itself divides into 200 once with remainder 8. That makes 8 such values. Thank you!
 
Last edited:
Thanks a lot for helping. Explanation is crystal clear.

Why was I thinking about common factors of 200 and 192 ? Please Get me rid of those feelings.
 
Quant Warrior said:
The remainder is 8 when 200 is divided by a positive integer 'x'. How many possible values of "x" are there ?
My efforts to solve this question:
I broke the number like this: 192 + 8. One thing that I concluded instantly is that 'x' will be larger than 8 since 8 is remainder. Now I have to find out all the factors of 192. The number of factors of 192 is 14. From 14, I subtracted common factors of 200 and 192. The number of common factors is 4(1,2,4,8). So the number of values that can divide 200 leaving remainder 8 is: 14-4=10.
Click to expand...
Why the he*l don't you use the available resources, LOOK HERE
 
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