If a and b are two-digit positive numbers and c is a three digit posit

Question

If a and b are two-digit positive numbers and c is a three digit positive number, such that c=a+b. Is the unit digit of c the same as the unit digit of a?

(1) All the digits of c are the same and all the digits of a are the same

(2) The tens digit of a is the same as the tens digit of b

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yashikaaggarwal · Accepted Answer

If a and b are two-digit positive numbers and c is a three digit positive number, such that c=a+b

Statement 1: All the digits of c are the same and all the digits of a are the same
C is a 3 digit no. Whose digits are same. Such as 111,222,333,444,555,666 and so on
Since C = A+B
Where A and B are two digit no.
The maximum value of C can be 99+99 = 198.
Therefore value of C has to be less than 222
And 111 is the only possible no.
Therefore C = 111
Since A is a two digit no. Whose both digits are same such as A = 11,22,33,........99
Now, 
Let Say, the unit digit of c the same as the unit digit of a
Therefore,
111=11+B
B = 100 (Not possible because B is a two digit number)
So A value can be any ranging from 22,33,44.....99
Which can never be equal to the unit digit of C.
(Sufficient)

Statement 2: The tens digit of a is the same as the tens digit of b
No constraint given to derive Unit place..
(Insufficient)

Answer is A

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RaghavKhanna · Answer

St. 1:  
If all digits of c are same and c = a+b
=>   c = 111  
=> Unit's digit of c = 1

Now, if all digits of 'a' are same
=>  'a' can take any value from these = {22,33,44,55,66,77,88,99}

=> Unit's digit of 'a' is never = 1

=>  St. 1 is sufficient

St. 2:  
Since, ten's digit of 'a' = ten's digit of 'b'
=>  Min value of ten's digit of 'a' as well as 'b' = 5 (Since, a+b>=100)

Now, if a = 53, b = 54
=> c = 117
=> Unit's digit of 'a' is not equal to unit's digit of 'c'

But, if a = 53, b = 50
=> c = 103
=> Unit's digit of 'a' is equal to unit's digit of 'c'

Therefore, st. 2 is insufficient.

=>  Answer: A

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • a and b are positive two-digit numbers.
 o a < 100 and b < 100 
• c is a three-digit positive number
• c = a + b ……………Eq.(i)
 o So, c < 100 + 100 ⟹ c < 200
o So, c must be in the form of 1xy, where x and y are single digit integers. 
• We need to find if the value of y is equal to the unit’s digit of a or not.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:   All the digits of c are the same and all the digits of a are the same
 • According to this statement: x = 1 and y = 1
 o Or c = 111 …….Eq.(ii) 
• Also, since all the digits of a are same, so possible value of a are { 11, 22, 33, 44, 55, 66, 77, 88, 99}
 o Now, unit’s digit of c and a will be equal only if a = 11, else they won’t be. So let’s check if a can be 11 or not.
  If a = 11, so from Eq.(i) and Eq.(ii), we have, b = c – a = 111-11 = 100
 • Which is not possible as b < 100   
• Hence, unit’s digit of c ≠ unit’s digit of a. 
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.

Statement 2:  The tens digit of a is the same as the tens digit of b
 • Let us assume that a = 10p + q, where p is a single-digit positive integer and q is a single digit non-negative integer.
• Thus, b must be in the form of b = 10p + r, where r is a non-negative single digit integer.
• Therefore, c = a + b = 20p + q + r.
 o Now, unit’s digit of c = ( 0 + q + r) = (q + r)
o However, we don’t know the value of r. 
  If r = 0 then unit’s digit of c = unit’s digit of a
 And if r ≠ 0, then unit’s digit of c ≠ unit’s digit of a  
• We are getting contradictory results. 
Hence statement 2 is not sufficient.
Thus, the correct answer is  Option A.

HoneyLemon · Answer

If a and b are two-digit positive numbers and c is a three digit positive number, such that c=a+b. Is the unit digit of c the same as the unit digit of a?

(1) All the digits of c are the same and all the digits of a are the same
 
SInce C = A + B 
c can never be 222 .

C must be 111 . 
S0 A can be anything from 22...99. (A , B are 2 digit numbers )
Hence A can never be 11 . 
 Sufficient

(2) The tens digit of a is the same as the tens digit of b
 
 It doesn't say anything about unit digit of C . So anything can be possible . 
 In-Sufficient

Ans is A .

bumpbot · Answer

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If a and b are two-digit positive numbers and c is a three digit posit

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