Around the World in 80 Questions (Day 7): When positive integer x is

Both conditions when put together will first give us values for x and y through which we can assess the possible values for m and n. Answer C

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When positive integer x is divided by 9, the remainder is m and when positive integer y is divided by 9, the remainder is n. If m > n, what is the value of m + n ?

x = 9p+ m
y= 9q +n
but m>n hence m cannot be 1 and n cannot be 8
hence m = 2,3,4,...,8
n = 1,2,3,...,7

(1) x + y is a multiple of 9.
multiple of 9 = 9,18,27
suppose x+y = 18
x=15,y=3 then m=6, n=3 hence m+n = 9
suppose x+y = 27
x=14, y = 13 then m=5,n = 4 m+n =9
suppose x=15, n = 12; m=6,n=3 m+n = 9

hence 1 is enough then the answer could be A or D
(2) x*y divided by 9, the remainder is 5
x*y= (9p+m)*(9q+n)=81pq+9pn+9qn+mn from this we know that m*n is 5
so m must be 5 and n must be 1 only then m+n =6 enough
answer is D

My Answer: E, both statements together are also insufficient.

Let, x=9a+m, y=9b+n

Statement (1): x-9a > y-9b
x-y > 9(a-b)

Also, 9a+9b+m+n is divisible by 9. So, m+n must be divisible by 9. Can't determine m+n from both these statements. Hence, insufficient.

Statement (2): xy=9c+5
(9a+m)(9b+m)=9c+5
Again can't determine m+n hence, insufficient.

Both statements together are also insufficient.

The answer is E
As per the stem x= 9q+m and y= 9q+n, we are asked to find the value of m+ n
As per statement 1 x+ y is a multiple of 9 which means that the sum of m and n will also be a multiple of 9. However finding a unique set of number for m+n is not possible as many numbers will qualify for the same. Hence insufficient.
As per statement 2 xy=9q + 5 is insufficient information too as we cannot deduce a unique value of m +n. Hence insufficient

m+n = ?

x = 9Q + m
y = 9z + n
m > n,, what is m + n?

1)(x+y) is a multiple of 9
=> 9Q + m + 9z + n = multiple of 9
=> m+n = multiple of 9 (m>n and both m and n are less than 9) ---> SUFFICIENT

2) xy = 9P + 5
=> (9Q + m)(9Z + n) = 9P + 5
=> 81QZ + 9Qn + 9Zm + mn = 9P + 5
=> 81QZ + 9Qn + 9Zm - 9P = 5 - mn
=> 9 (9QZ + Qn + Zm - P) = 5 - mn ---> NOT SUFFICIENT

Answer - [A]

x = 9j + m
y = 9k + n
m>n are given

1. x + y is a multiple of 9.
x+y = 9*(j+k) + (m+n)
For x+y to be multiple of 9, m+n must be divisible by 9.
1<=m<=8
1<=n<=8
3<=m+n<=15 because m>n
The only multiple of 9 in this range is 9 itself. So, m+n is 9.
Hence, 1 is sufficient.

2. x*y divided by 9, the remainder is 5
(9j + m)(9k+n) = 9p+5
The only term with no 9 is mn
mn must be of the form 9a+5, where a is a non-negative integer (including 0).
m= 5, n =1 is a possible solution, which gives m+n = 6
m=7, n=2 is another solution, which gives m+n = 9
So, a unique solution is not possible.
Hence 2 is not sufficient.

A is the answer.

x = 9p + m
y = 9q + n
m>n
x & y are +ve integers

Value of m+n?

1) x + y is a multiple of 9

Adding both the equations mentioned above

x+y = 9p + 9q + m + n
= 9(some integer) + m + n

If x+y is a multiple of 9, that means m+n is also a multiple of 9

Since m & n < 9

That means m+n = 9

Sufficient

2) x*y divided by 9, the remainder is 5

xy = (9p+m) * (9q+n)
= 9(some integer) + mn

Now the remainder is 5 but it doesn't mean that mn = 5

Since m>n, m can be 5 and n can be 1 making mn = 5 and m+n = 6

m can also be 7 and n can be 2 making mn = 14 which when divided by 9 gives remainder of 5. In this case m+n = 9

Not sufficient

Ans A

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given
x=9a+m
y=9b+n
if m>n , value of m+n?
#1
x + y is a multiple of 9.
x+y can be 9,18,27,36,45

17=9*1+8
1=9*0 +1
m+n=9
x+y=27

6=9*(0)+6
21=9*+3
m+n=9

7=9*0+7
20=9*2+2
m+n=9

x+y=36
30=9*3+3
6=9*0+6
m+n=9

15=9*1+6
21=9*2+3

sufficient that m+n is 9

#2
x*y divided by 9, the remainder is 5
possible values of x*y will be
1*5 ; 2*7 ; 1*23 ; 2*16 ; 1*41 ; 2*25
5=9*0+5
1=9*0+1
m+n=6

16=9*1+7
2=9*0+2
m+n=9

41=9*4+5
1=9*0+1

25=9*2+7
2=9*0+2

we get two values of m+n i.e. 6 & 9

insufficient



[b]OPTION A is correct[/b]

Given: When positive integer x is divided by 9, the remainder is m and when positive integer y is divided by 9, the remainder is n.
Asked: If m > n, what is the value of m + n ?

x = 9k + m; k is an integer; 0<m<9
y = 9l + n; l is an integer; 0<n<9
m > n
m + n = ?

(1) x + y is a multiple of 9.
x + y = 9k + m + 9l + n = 9(k+l) + (m+n)
Since 0<m<9 & 0<n<9; 0<m+n<18
Since x + y is a multiple of 9; m+n is a multiple of 9
m + n = 9
SUFFICIENT

(2) x*y divided by 9, the remainder is 5
x*y = (9k + m ) (9l + n) = 81kl + 9lm + 9kn + mn = 9(9kl + lm + kn) + mn
0 < mn < 9*9 = 81
Value of m+n can not be ascertained.
NOT SUFFICIENT

IMO A

We know that m, n are integers
(1) x+y is divisible by 9 -> m+n must be 9 -> Sufficient
(2) x*y will have remainder m*n which is 5. Since m*n=5 and m,n are integers, m and n must be 1 and 5 or 5 and 1 -> m+n=6 -> Sufficient

Answer D

When positive integer x is divided by 9, the remainder is m and when positive integer y is divided by 9, the remainder is n. If m > n, what is the value of m + n ?

(1) x + y is a multiple of 9.
(2) x*y divided by 9, the remainder is 5
__________________________________________

x/9 = .. (m remainder)
y/9= ... (n remainder)
If m>n , what is m+n ?
m can be maximum 8 and c can be max 7 or less.

(1) is not sufficient , as the remainders could be any number between 9 and 0. THere is no restriction.
(2) when x*y is multiplied. It gives us a product lets say X. When we divide the X by 9, it gives us remainder of any number. thus it is not sufficient.

Combination helps us to find the answer.
C may be our winner.

Let x = 9k + m and y = 9l + n, where k and l are integers and 8 >= m > n >= 0 (since m and n are remainders when divided by 9).
To find, m+n = ?

(1) x + y is multiple of 9.
x + y = 9(k+l) + (m+n)
so m+n = multiple of 9 = 9, 18...
As 8 >= m > n >= 0 , m+n =< 16 i.e. m+n = 9. Sufficient.

(2) x*y divided by 9, the remainder is 5
x*y = 81kl + 9kn + 9lm + mn
Remainder when x*y divided by 9 = Remainder when mn divided by 9.
Possible values of mn = 5, 14, 23, 32 etc.
So (m,n) = (5,1), (7,2), (8,4) etc.
So m+n can not be found out. Insufficient.

Only (1) is sufficient.

IMO A.

When positive integer x is divided by 9, the remainder is m and when positive integer y is divided by 9, the remainder is n. If m > n, what is the value of m + n ?



(1) x + y is a multiple of 9.
so the values of (x,y) = (8,1), (7,2), (6,3), (5,4) and (17,1), (16,2), (15,3), so on and so forth, it looks like the remainder will be the same if we take the values from (1 to 9) or 10 to 18 or 19 to 27, so on
Corresponding values of (m,n)since m>n = (8,1), (7,2), (6,3), (5,4)
hence m+n = 9
Hence sufficient.
(2) x*y divided by 9, the remainder is 5
so (x,y) =(5*1), (7,2), (23,1),(16,2), (8,4) etc.
corresponding (m,n) =(5,1), (7,2), (5,1), (7,2) etc.
so sum m+n = 6 or 9
Hence not sufficient.

So A will be my answer.

x=9k+m
y=9t+n
1) Adding both above equation will tell us that m+n has to be equal to 9. Please note that they can't be more than 9 otherwise the equation will not hold true. Sufficient.
2) This condition is not sufficient as it will only give multiple of reminders. It does not give details about their sum. Insufficient.
Hence answer is A

When positive integer x is divided by 9, the remainder is m and when positive integer y is divided by 9, the remainder is n. If m > n, what is the value of m + n ?

(1) x + y is a multiple of 9.
(2) x*y divided by 9, the remainder is 5

Answer:
positive integer x is divided by 9, the remainder is m
i.e x= 9j +m

positive integer y is divided by 9, the remainder is n
i.e y = 9k +n

m>n, so m+n?


Case I: x + y is a multiple of 9, which means x+y is divisible by 9
Adding x and y from question
we get x+y = 9 (j+k) +m+n
so m+n is divided by 9 as it is completely divisible
so regardless of m>m, m+n = 9
hence this option is sufficient

Case II: x*y divided by 9 , remainder is 5
xy=(9j+m)(9k+n)= 81jk +9j*n +9k*m + mn
No when x*y is divided by 9
each of the term gets divided by 9 except mn
so mn = 5
and m >n
and m and n are integers
so possible options are m = 5 and n =1
so m + n =6
so this option is sufficient

Hence D is the answer

When positive integer x is divided by 9, the remainder is m and when positive integer y is divided by 9, the remainder is n. If m > n, what is the value of m + n ?
=>
x= 9k1+m
y=9k2+n
(1) x + y is a multiple of 9.
x+y = 9(k1+k2)+m+n
now m+n = 9k3
it can take any value so not sufficient
(2) x*y divided by 9, the remainder is 5
x*y = 81k1k2 + 9k1n + 9k2m + mn
mn = 5
5*1 or 1*5
m+n = 6
so sufficient

hence ans is B

given \(x/9 = m\) means m<9
similarly m]y/9 = n[/m] means n<9
m > n
m + n ?

stmt 1
we know that \((x + y)/9 = rem 0\)
then \((m + n)/9 = rem 0\)
also both m and n are less then 9 and m>n
then only possible value of m + n = 9
sufficient

stmt 2
we know that \((x * y)/9 = rem 5\)
then \((m * n)/9 = rem 5\)
possible values of m*n = 5, 14, 23, 32.........
but both m and n are less then 9 and m>n
then possible values of m and n are (5,1) and (7,2)
in both cases value of m + n is different hence insufficient

correct answer A