Around the World in 80 Questions (Day 10): If m and n are positive

Answer E

We cannot find the remainder unless we know the value of m*n for sure for which none of the statements seem sufficient

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given that m& n are +ve integers
target what is remainder when m*n is divided by 49
#1
m - n is divisible by 42.
(43,1) ; ( 50-8) ; (55-13)
remainder values are different
insufficient
#2
m*n is divisible by 35.
m*n has to be a factor of 35
35,1 ; 35,2 ; 70,1 ; 105,1 ; 140,1

again remainder values are different
insufficient
from 1 &2
m,n has to be a factor of 35 such that least value is
70,28 ; 105,63 ; 140,98
all these pair values are factors of 7 and the remainder here when m*n is divided by 7^2 will be 0
sufficient
option C

(1) m - n is divisible by 42.

A) m = n = 42*49

what is the reminder when m*n is divided by 49 ? - The remainder = 0

B) m = 43, n = 1

what is the reminder when m*n is divided by 49 ? - The remainder = 43

Not sufficient.

(2) m*n is divisible by 35.

A) m = n = 35*49

what is the reminder when m*n is divided by 49 ? - The remainder = 0

B) m = 7, n = 5

what is the reminder when m*n is divided by 49 ? - The remainder = 35

Not sufficient.

Combined

m * n is divisible by 35, so either m or n has 7 as its prime factor
Combined with
m - n is divisible by 42, so both m and n must have 7 as its prime factor.

Hence, m * n will have two seven in its prime factorized form ; m*n will be divisible by 49.

Sufficient.

IMO C

The Answer is C
we need to find the value of the remainder when mXn is divided by 49
Statement A says that m - n is divisible by 42. Many values will fulfil this criteria(m-n) Such as 1, 2,3, 6,7, 14, 21. They all will give different remainders on being divided by 49. Hence insufficient
Statement B says that mxn is divisible by 35, Many values satisfy this equation such as 1,5,7, They all will give different values on being divided by 49
Combining both the only value where m-n is divisible by 42 and mxn is divisible by 35 is 7, which is divisible by 49.
Hence C is sufficient.

statement 1
divisible by 42
m-n can be 1,2 3 ,6 ,7 ,14,21 ,42
we cannot find m*n from only this data
statement 2
m*n is divisible by 35
(m,n)can be 5,7 or 7,5 or 35,1 or 1,35
all possible products are 35
so m*n is equals to 35
so remainder will be 49/35=14 as remainder

If m and n are positive integers, what is the reminder when m*n is divided by 49 ?

(1) m - n is divisible by 42.
(2) m*n is divisible by 35.
_____________________________
both M and N are positive integers.
(1) divisable by 42 means, it is also divisable by 7 and 6.
if M=85 ; N =43 , it can be divided by 42 or M=84 ; N=42, Still it is insufficient.
(2) mn can be 35 or lets say 70 . In this case , remainder will be different.

when we combine both statements, it could help us to solve.
C is our answer.

(2)

If m and n are positive integers, what is the reminder when m*n is divided by 49 ?

(1) m - n is divisible by 42.
(2) m*n is divisible by 35.

Case I: m-n is divisible by 42
so if m =43 and n=1 , m*n is not divisible by 49
however if m =49 and n =7, m-n becomes 42 and is divisible by 42
so we do not get unique answer from the same
Hence this option is not sufficient

Case II: m*n is divisible by 35
which means m can be 35 , n can 1....in this case mn is not divided by 49
or m can be 7 and n can be 5,....... in this case mn is not divided by 49
or m can be 35 and n can be 7.....in this case mn is divided by 49
so again we do not get unique answer from the same
Hence this option is not sufficient

Combining both,
m*n is divisible by 35 means it is divisible 7 and 5 so either one of the two would be a multiple of 5 and other a multiple of 7 or limiting condition m=35 and n=1 but that does not satifsy m-n=42
m-n=42 which means both differ by 42 units which is a multiple of 7
so combining m*n we get conclusive answer that mn is divisible by 49

C is the answer

S1:- Let m= 50 & n = 8, then m-n will be divisible by 42
Now remainder when 50*8 is divided by 49 = 8

Now let m = 51 & n = 9. Again m-n is divisible by 42
Remainder when 51*9 is divided by 49 = 18

No unique solution. S1 not sufficient.


S2:- Let m= 7 & n = 5, then m*n will be divisible by 35
Now remainder when 7*5 is divided by 42 = 35

Now let m = 14 & n = 10. Again m*n is divisible by 35
Remainder when 14*10 is divided by 49 = 42

No unique solution. S2 not sufficient.

S1 & S2 together :- For m*n to be divisible by 35, it should be a multiple of 5 and 7. And also, m & n have to be chosen for which the difference of these two will be 42. To be a multiple of 5, last digit should be either 0 or 5, and to maintain a difference of 42, the last digit of second number should be 8 or 3. Such condition would exist only when m and n both are factors of 7. So, m*n will be a factor of 49 and remainder will be 0.

e.g. - Let m= 70 & n = 28, then m*n will be divisible by 35 and m-n will be divisible by 42
Now remainder when 105*63 is divided by 49 = 0

Hence correct choice is C.

If m and n are positive integers, what is the reminder when m*n is divided by 49 ?

(1) m - n is divisible by 42
This statement is insufficient as it does not provided adequate information.
(2) m*n is divisible by 35.
This statement is insufficient as it does not provided adequate information

Both statements together are insufficient as they do not provided adequate information
Answer = E

Answer is E
If m and n are positive integers, what is the reminder when m*n is divided by 49 ?

(1) m - n is divisible by 42.
we know that m & n both have factors of 2, 3, and 7; Insufficient
(2) m*n is divisible by 35.
We know that m or n has or both have factors of 5, and 7. Insufficient.
(1)+(2) still insufficient.

If m and n are positive integers, what is the reminder when m*n is divided by 49 ?

(1) m - n is divisible by 42.
let m = 82, n = 40, 82*40 / 49 , remainder = 7
let m = 71, n = 31, 71 *31/ 49, remainder = 45
not sufficient

(2) m*n is divisible by 35.
mn = 35 , 35/49, remainder = 35
mn = 70, 70/49 , remainder = 49
not sufficient

Lets try C , mn is multiple of 35 and difference between m-n is 42
multiple of 35 will always result in units digit of 5, or 0, in order to get 42, other number has to have unit digit of 8 (which is not a multiple of 35)
so the remainder will be 0, sufficient

Answer: C

Statement (1) tells us that m - n is divisible by 42. However, this information alone does not provide enough details about the values of m and n to determine the remainder when m*n is divided by 49.

Statement (2) informs us that m*n is divisible by 35, meaning both m and n are divisible by 5 and 7. Since 35 is divisible by 7, we can conclude that the remainder when m*n is divided by 49 is 0.

Since Statement (2) gives us a definitive answer for the remainder, it is sufficient to solve the problem. Hence, the correct answer is (B) Statement (2) alone is sufficient.

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If m and n are positive integers, what is the reminder when m*n is divided by 49 ?

Prework:
Say m*n/49 gives remainder r with quotient q.
49q + r = mn
r = mn - 49q .(i)
We have to find the value of r.

(1) m - n is divisible by 42.
Doesn't help. Insufficient

(2) m*n is divisible by 35.
mn/35 = y
mn = 35y
substitute in (i)
r = 35y-49q
r = 7(5y-7q)
r is divisible by 7. We do not know about 49. Insufficient

Thus answer E.

S1) m-n=42,84...
take m=43, n=1, m*n leaves remainder as 43 when divided by 49

Take m=44, n=2, m*n=88 leaves remainder of 39 when divided by 49
(NS)

S2) m*n=35,70...
Rem(35/49)=35, Rem(70/49)=31
(NS)

S1+S2) The multiples of 35 is 35(35+42), 70(70+42)... all give remainder of 0 when div by 49 .

Hence Suff

C)

In order to find the reminder, we need multiples.
1) m - n is divisible by 42
This does not tell us anything about the value of their multiples. Insufficient.
2) m*n is divisible by 35
This only tells us that m*n is 35k, where k is an integer. Reminder will also depend upon k hence insufficient.

Taking together, we don't get any information on k. Hence, together insufficient. E is the answer.

St1.
I. m-n=42k
43-1=42

43/49
Reminder= 43

II. 85-1=42*2

85/49
Reminder=36

Insufficient

St2.
I. m*n=35k
35*1=35*1

35/49
Reminder=35

II. 70*1=35*2

70/49
Reminder=21

Insufficient

St1 and St2
We can take the LCM of both 42 and 35. Still we get different reminder.

E

To find: The remainder when m*n is divided by 49?

Statement 1: m - n is divisible by 42

If m=44,n=2
mn/49 r=39

If m=49,n=7
mn/49 r=0

Statement 1 is not sufficient.

Statement 2: m*n is divisible by 35

If m=35,n=1
mn/49 r=5

If m=70,n=1
mn/49 r=3

Statement 2 is not sufficient.

Combining statements 1 and 2 is not sufficient as,

If m=44,n=2
m-n is divisible by 42 but mn is not divisible by 35.


Answer is E.

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