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Set P consists of five distinct positive integers {a, b, c, d, e}. The median of the set is M and its standard deviation is integer D such that 6≤D≤9. When M is divided by D, the remainder is 5. When a positive integer x is added to all the numbers of the set P, the new median of the set P is M’ and the new standard deviation is D’. If x leaves a remainder not less than 8 when divided by D’ and M’ leaves a remainder r when divided by D’, by how much is r less than or greater than the remainder obtained when M was divided by D?
A. 20% less
B. 20% greater
C. 25% less
D. 60% greater
E. 160% greater
A. 20% less
B. 20% greater
C. 25% less
D. 60% greater
E. 160% greater
07 Dec 2023, 06:51
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Beautiful Problem !!
Let us assume that the set is in ascending order, i,e, a<b<c<d<e.
Therefore, median of the set M = c
From the problem we get,
\(Remainder of (c/D) = 5 ......................(I)\\
\)
D' is really a dummy because the standard deviation does not change when the same number is added to all the elements of the set.
Therefore, D'=D
Since, all elements are added by x, the Median of the new set becomes (c+x).
\(Remainder of (x/D) >=8\) (from the problem statement) .......................(II)
\(Remainder of [(c+x)/D] = r\) (from the problem statement) ......................(III)
If we see statement II, D cannot be less than or equal to 8 because if D is less than or equal to 8, it cannot leave a remainder that is greater or equal to 8. Since the divisor is always greater than the remainder, we can conclude that,
\(D>8\)
But in the problem statement we are told that 6<=D<=9.
So combining the two statements, we can see that the only possible value of D is 9.
Again see statement (II), which has now turned into:
\(Remainder of (x/9) >=8\)
But realise that remainder of (x/9) must always be less than 9 (Because if it becomes 9, the quotient increases by 1 and the remainder becomes zero).
So, from the above two statements, we conclude:
\(Remainder of (x/9) =8 ......................(IV) \)
Recall statement (III) :
\(Remainder of [(c+x)/D] = r \)(from the problem statement) ......................(III)
We use a little property of modular arithmetic here which states (c+x)mod D = ((c mod D) + (x mod D))mod D
which put in simple terms, translates to:
\(Remainder of [(c+x)/D] = Remainder of [(Remainder of c/D) + Remainder of (x/D))/D]\)
Put D=9 in the equation:
\(Remainder of [(Remainder of c/9) + Remainder of (x/9))/9]\)
We know from statement (I) and (IV), the values of (Remainder of c/9)=5 and Remainder of (x/9) = 8,
=> r = Remainder of [(Remainder of c/9) + Remainder of (x/9))/9] = Remainder of [(5+8)/9] = Remainder of (13/9) = 4.
So, when M was divided by D, remainder was 5 (Statement I).
r=4
Therefore r is 20% smaller than 5.
Option A.
Let us assume that the set is in ascending order, i,e, a<b<c<d<e.
Therefore, median of the set M = c
From the problem we get,
\(Remainder of (c/D) = 5 ......................(I)\\
\)
D' is really a dummy because the standard deviation does not change when the same number is added to all the elements of the set.
Therefore, D'=D
Since, all elements are added by x, the Median of the new set becomes (c+x).
\(Remainder of (x/D) >=8\) (from the problem statement) .......................(II)
\(Remainder of [(c+x)/D] = r\) (from the problem statement) ......................(III)
If we see statement II, D cannot be less than or equal to 8 because if D is less than or equal to 8, it cannot leave a remainder that is greater or equal to 8. Since the divisor is always greater than the remainder, we can conclude that,
\(D>8\)
But in the problem statement we are told that 6<=D<=9.
So combining the two statements, we can see that the only possible value of D is 9.
Again see statement (II), which has now turned into:
\(Remainder of (x/9) >=8\)
But realise that remainder of (x/9) must always be less than 9 (Because if it becomes 9, the quotient increases by 1 and the remainder becomes zero).
So, from the above two statements, we conclude:
\(Remainder of (x/9) =8 ......................(IV) \)
Recall statement (III) :
\(Remainder of [(c+x)/D] = r \)(from the problem statement) ......................(III)
We use a little property of modular arithmetic here which states (c+x)mod D = ((c mod D) + (x mod D))mod D
which put in simple terms, translates to:
\(Remainder of [(c+x)/D] = Remainder of [(Remainder of c/D) + Remainder of (x/D))/D]\)
Put D=9 in the equation:
\(Remainder of [(Remainder of c/9) + Remainder of (x/9))/9]\)
We know from statement (I) and (IV), the values of (Remainder of c/9)=5 and Remainder of (x/9) = 8,
=> r = Remainder of [(Remainder of c/9) + Remainder of (x/9))/9] = Remainder of [(5+8)/9] = Remainder of (13/9) = 4.
So, when M was divided by D, remainder was 5 (Statement I).
r=4
Therefore r is 20% smaller than 5.
Option A.
07 Dec 2023, 06:59
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jitmitra2005
A more notation friendly way to do it is really using modular arithmetic.
Given:
c mod D = 5 ..... (I)
x mod D >=8 .....(II)
(c+x) mod D = r ......(III)
From II, D>8
From problem statement, 6<=D<=9.
Combining above two statements, D=9
Again see statement (II), we use the property, x mod D < D,
So, x mod 9 < 9
Therefore, x mod 9 = 8
Use statement III,
(c+x) mod D = r
=> ((c mod D) + (x mod D))mod D = r
=> ((c mod 9) + (x mod 9))mod 9 = r
Use results above:
r = (5+8)mod 9 = 13 mod 9 = 4
Therefore, r=4.
4 is 20% less than 5. Option A.
