If x and y are two digits and the number $4x5y790$ is divisible by 11, then what is the remainder, if $x+y$ is divided by 11?

The correct option is (d) - 7.

[as per provisional answerkey]

Solution

To determine the divisibility of a number by 11, we use the divisibility rule: The difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11.

Given number: $4x5y790$
Positions (from left to right):
1st: 4, 2nd: x, 3rd: 5, 4th: y, 5th: 7, 6th: 9, 7th: 0

Step 1: Calculate the sums of digits at odd and even positions.
Sum of digits at odd positions (S1) = $4+5+7+0=16$
Sum of digits at even positions (S2) = $x+y+9$

Step 2: Apply the divisibility rule.
The difference $|S1-S2|$ must be 0, 11, 22, etc.
Difference = $|16-(x+y+9)|=|7-(x+y)|$

Step 3: Solve for $(x+y)$.
Case 1: $7-(x+y)=0$
$x+y=7$
Case 2: $7-(x+y)=-11$ (since x and y are digits, x+y cannot be large enough to make the difference -22)
$x+y=7+11=18$
Case 3: $7-(x+y)=11$
$x+y=-4$ (Not possible as x and y are digits)

Step 4: Find the remainder when $(x+y)$ is divided by 11.
If $x+y=7$, then $7/11$ gives a remainder of 7.
If $x+y=1$8, then $18/11$ gives a remainder of 7 ($18=11\times 1+7$).
In both possible cases, the remainder is 7.

Why the other options are incorrect

• Option (a) - 1: This would require $(x+y)$ to be 1 or 12. If $x+y=1$, the difference $|7-1|=6$ (not divisible by 11). If $x+y=12$, the difference $|7-12|=5$ (not divisible by 11).
• Option (b) - 3: This would require $(x+y)$ to be 3 or 14. If $x+y=3$, the difference $|7-3|=4$. If $x+y=14$, the difference $|7-14|=7$. Neither is divisible by 11.
• Option (c) - 5: This would require $(x+y)$ to be 5 or 16. If $x+y=5$, the difference $|7-5|=2$. If $x+y=16$, the difference $|7-16|=9$. Neither is divisible by 11.

Key Concept

The Divisibility Rule of 11 states that the difference between the sum of digits in odd places and even places must be a multiple of 11 (including 0).