Around the World in 80 Questions (Day 3): If x and y are consecutive

Question

If x and y are consecutive positive two-digit integers, what is the value of x + y?

(1) When x^25 is divided by y the remainder is 11
(2) When x^36 is divided by y the remainder is 1

missionmba2025 · Accepted Answer

An interesting thing to note down here is x and y are consecutive positive two-digit integers. So the remainder of (x/y) is either +1 or -1 depending on the relative position of x and y.

(1) When x^25 is divided by y the remainder is 11

Say x/y were +1, the the remainder of x^25 divided by y would be 1. But that's not the case, the remainder is -1. Hence x < y, and x = 11, y =12. We can find the sum of x and y.

Sufficient

(2) When x^36 is divided by y the remainder is 1

Whether the remainder of x/y is +1 or -1, the remainder of x^36/y will always be +1.

Hence, x and y can be any two digit consecutive numbers

Not sufficient.

IMO A

Archit3110 · Answer

given that x & y are consecutive two digit integers
value of x+y
#1
When x^25 is divided by y the remainder is 11
for every odd power of x the remainder will be same as value of x i.e. here x is 11 and y is 12 ; x+y is 23
sufficient
#2
When x^36 is divided by y the remainder is 1
for every even power of x the remainder is always 1 for all consecutive integers 
we do not get single value of x,y
insufficient
 
 OPTION A is correct

KanikaJain · Answer

If x and y are consecutive positive two-digit integers, what is the value of x + y?

(1) When x^25 is divided by y the remainder is 11
(2) When x^36 is divided by y the remainder is 1

case I:
x^25 = y*a + 11
thus y = 12, 13....
now if y = 12, x = 11 or 13
x = 11. thus 11^25/12 remainder is -1 = 11
x = 13. thus 13^25/12 remainder is 1

now if y = 13, x = 12 or 14
x = 12. thus 12^25/13 remainder is -1 = 12
x = 14. thus 14^25/13 remainder is 1

case II:
x^36 = b* y + 1
thus if y = 12, x = 11 or 13
x = 11^36/12, remainder = 1
x = 13^36/12, remainder = 1

thus if y = 11, x = 10 or 12
x = 10^36/11, remainder = 1
x = 12^36/11, remainder = 1

thus x= 11, y = 12

thus A

Theprince · Answer

here there are 2 possiblities of x and y ( they are consecutive int )
if x>y then x raised to anything divided by y will give us 1 as remainder
and if x<y then x raised to anything divided by y will give us -1 as remainder
I found this link to be helpful :- https://prepinsta.com/how-to-find-remai ... s-quickly/

1. when x^25 is divided by y then remainder is 11 now this case we have 2nd condition
hence -1 will translate to y = 12 and x should be less than y hence x= 11
suff

2. when x^36 is divided by y then remainder is 1.
here we can have any 2 consecutive int x and y such that x>y we will get remainder 1
insuff

Ans A

UtkarshAnand · Answer

If x and y are consecutive positive two-digit integers, what is the value of x + y?

(1) When x^25 is divided by y the remainder is 11

Only option is x=11, and y =12

11^25 mod 12
= (-1)^25 mod 12 (Note :- 11 mod 12 = 11 (or, -1 in terms of -ve remainder))
= (-1) mod 12
= -1
= -1+12
= 11

So, x+y =11 +12 =23

Hence, SUFFICIENT

(2) When x^36 is divided by y the remainder is 1

11^36 mod 12 (x=11, y =12)
= (-1)^36 mod 12 (Note :- 11 mod 12 = 11 (or, -1 in terms of -ve remainder))
= 1 mod 12
= 1
So, x+y =11 +12 =23......(1)

Also, if
12^36 mod 13 (x=12, y =13)
= (-1)^36 mod 13 (Note :- 12 mod 13 = 12 (or, -1 in terms of -ve remainder))
= 1 mod 12
= 1
So, x+y =12 +13=25....(2)

Since, two values possible (as above) for x+y

Hence, INSUFFICIENT

(A) is the CORRECT answer

pintukr · Answer

If x and y are consecutive positive two-digit integers, what is the value of x + y?

(1) When x^25 is divided by y the remainder is 11

Only option is x=11, and y =12

11^25 mod 12 
= (-1)^25 mod 12 (Note :- 11 mod 12 = 11 (or, -1 in terms of -ve remainder))
= (-1) mod 12
= -1
= -1+12 
= 11

So, x+y =11 +12 =23

Hence, SUFFICIENT

(2) When x^36 is divided by y the remainder is 1

11^36 mod 12 (x=11, y =12)
= (-1)^36 mod 12 (Note :- 11 mod 12 = 11 (or, -1 in terms of -ve remainder))
= 1 mod 12
= 1
So, x+y =11 +12 =23......(1)

Also, if
12^36 mod 13 (x=12, y =13)
= (-1)^36 mod 13 (Note :- 12 mod 13 = 12 (or, -1 in terms of -ve remainder))
= 1 mod 12
= 1
So, x+y =12 +13=25....(2)

Since, two values possible (as above) for x+y

Hence, INSUFFICIENT

(A) is the CORRECT answer

bumpbot · Answer

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Around the World in 80 Questions (Day 3): If x and y are consecutive

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