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23 Sep 2022, 07:14
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D
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Difficulty:
95%
(hard)
Question Stats:
39% (02:14) correct
61%
(02:22)
wrong
based on 98
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Five different integers are selected at random from the numbers 1 to 100. Each of the selected numbers is then divided by integer n. What is the value of n ?
(1) The greatest possible remainder for any selected integer is 4.
(2) For some selected integer, k, the quotient when k is divided by n is 9, the quotient when k is divided by n + 2 is 6, and the difference of the remainders is 3.
(1) The greatest possible remainder for any selected integer is 4.
(2) For some selected integer, k, the quotient when k is divided by n is 9, the quotient when k is divided by n + 2 is 6, and the difference of the remainders is 3.
23 Sep 2022, 10:15
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We need to know what is the value of Integer n.
(1) The greatest possible remainder for any selected integer is 4.
=> If an Integer is divided by 5 the possible remainders are 0,1,2,3,4. Hence, we know n = 5
Statement 1 alone is sufficient.
(2) For some selected integer, k, the quotient when k is divided by n is 9, the quotient when k is divided by n + 2 is 6, and the difference of the remainders is 3.
=> Let remainder when k is divided by n = a and remainder when k is divided by 6 = b
k = 9n + a and k = 6(n+2)+b
=> 9n+a = 6n+12+b
=> 3n = 12 + (b-a)
Also, given difference of remainders = 3, but we don't know whether a>b
=> So, either b-a=3 or a-b=3
=> 3n = 15 or 3n = 9
i.e., 'n' could be 3 or 5.
e.g., Statement 2 is valid in both cases:
if n=3 and k=30 (a=3;b =0) or if n=5 and k=45 (a=0; b=3)
Statement 2 alone is not sufficient.
Answer: A
(1) The greatest possible remainder for any selected integer is 4.
=> If an Integer is divided by 5 the possible remainders are 0,1,2,3,4. Hence, we know n = 5
Statement 1 alone is sufficient.
(2) For some selected integer, k, the quotient when k is divided by n is 9, the quotient when k is divided by n + 2 is 6, and the difference of the remainders is 3.
=> Let remainder when k is divided by n = a and remainder when k is divided by 6 = b
k = 9n + a and k = 6(n+2)+b
=> 9n+a = 6n+12+b
=> 3n = 12 + (b-a)
Also, given difference of remainders = 3, but we don't know whether a>b
=> So, either b-a=3 or a-b=3
=> 3n = 15 or 3n = 9
i.e., 'n' could be 3 or 5.
e.g., Statement 2 is valid in both cases:
if n=3 and k=30 (a=3;b =0) or if n=5 and k=45 (a=0; b=3)
Statement 2 alone is not sufficient.
Answer: A
30 Sep 2022, 11:09
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Aks111
I noticed where I went wrong. The difference of remainders is 3. If n=3 we cannot have remainder as 0 and 3.
The maximum difference of remainders when n=3 is 2.
Statement 2 alone is sufficient as we get unique value for n (n=5).
Answer: D
