p, q, r, s are positive integers. When p is divided by q, the

Question

p, q, r, s are positive integers. When p is divided by q, the remainder is 7; when r is divided by s, the remainder is 12. Which of the following may be a value of q + s?

A. 15
B. 17
C. 19
D. 20
E. 23

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vasu1104 · Answer

p = qD + 7 
 r = sM + 12

here D, M are dividend for each equation respectively. 
 now if we take any number, the maximum remainder we can is one number less than that number. 
 say the divisor is 7. the maximum remainder we can get is 6. when 13 divided by 7, the remainder is 6. but when 14 is divided by 7, the remainder is 0. so the cycles repeats after getting remainder 6.

so to get remainder 7, we need to divide by 8. 
 say 15 is divided by 8, the remainder is 7. 
 but for 16, its 0. 
 then to divide 23, it again 7.

so we found that q= 8

to get 12 as remainder the number could be 13 or 15 either.

but when we look at the choice we need to go with s=15

r+s= 8+15 =23

shivani1351 · Answer

Rule for Remainders: The divisor must always be greater than the remainder.
As the remainder is 7, the divisor q must be greater than 7. Thus, the smallest possible value for q = 8
Also, as the remainder is 12, the divisor s must be greater than 12. Thus, the smallest possible value for s = 13
Hence, the smallest possible value of q+s = 8+13 = 21

Since all the other options except Option E are less than 21, we can eliminate them. Hence, the correct answer is Option E - 23

Hope this helps!

ExpertsGlobal5 · Answer

Video explanation:

https://www.youtube.com/watch?v=FUd1tuGt7_k

himssss29 · Answer

When dividing, remainder must be  less than divisor: 
 
 Remainder 7 → q > 7 →  q ≥ 8  
 Remainder 12 → s > 12 →  s ≥ 13   
Minimum q + s:

q+s≥ 8+13=21
 answer above 21 is option E.

quaeneque · Answer

when 27 i divded by 15, remainder is 12, 15 + 8 = 23 ( >= 13 + 8 >= 21)

p, q, r, s are positive integers. When p is divided by q, the

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p, q, r, s are positive integers. When p is divided by q, the

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