Question

If the 9 digit number 807x6y9z8 is divisible by 99, the value of √x+y+z?
1. 6
2. 5
3. 4
4. 3√3​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

If the 9 digit number 807x6y9z8 is divisible by 99, the value of √x+y+z

1. 6

2. 5

3. 4

4. 3√3

CONCEPT TO BE IMPLEMENTED

Divisibility by 9 :

A number is divisible by 9 if the sum of digits is divisible by 9.

Divisibility by 11 :

A number is divisible by 11 if the difference of the sum of the digits at even places and the sum of the digits at odd places is divisible by 11.

EVALUATION

Here the given number is 807x6y9z8

Since the number is divisible by 99

So the number is divisible by both 9 and 11

As 807x6y9z8 is divisible by 9

So sum of digits is divisible by 9

Sum of the digits = 38 + x + y + z

⇒ 38 + x + y + z is divisible by 9 - - - (1)

Again 807x6y9z8 is divisible by 11

Then the difference of the sum of the digits at even places and the sum of the digits at odd places is divisible by 11.

Thus

( 0 + x + y + z ) - ( 8 + 7 + 6 + 9 + 8 ) is divisible by 11

⇒ x + y + z - 38 is divisible by 11 - - - - (2)

(1) and (2) holds when x + y + z = 16

Thus we get

$\sf{ \sqrt{x + y + z} = \sqrt{16} = 4 }$

FINAL ANSWER

Hence the correct option is 3. 4

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Answer 2

Solution :-

Since , 807x6y9z8 is divisible by 99 = 9 * 11 . Then it must be divisible by both 9 and 11.

we know that,

• if sum of all digits is divisible by 9, then the number also divisible by 9.
• A number to be divisible by 11 , the difference between the sum of the odd numbered digits (1st, 3rd, 5th...) and the sum of the even numbered digits (2nd, 4th...) must be divisible by 11.

So,

By Divisibility Rule of 9 we get :-

→ (8+0+7+x+6+y+9+z+8) ÷ 9

→ (38 + x + y + z) ÷ 9

Conclusion 1) :- So, Possible values of (x + y + z) are :-

• 45 - 38 = 7
• 54 - 38 = 16
• 63 - 18 = 25

Conclusion 2) :- Now, by Divisibility Rule of 11 now, we get :-

→ { (0 + x + y + z ) - (8 + 7 + 6 + 9 + 8) } = 0, 11, 22, 33 ____

→ (x + y + z) - (38) = 0, 11(Or,-11), 22(oOr,-22), 33(Or,-33) ____

therefore, From Both Conclusions , we get ,

• 7 - 38 = (-31) ≠ Not divisible by 11 .
• 16 - 38 = (-22) = Divisible by 11 .
• 25 - 38 = (-13) ≠ Not divisible by 11 .

hence, we can conclude that, x + y + z is equal to 16 .

So,

→ √(x + y + z)

→ √(16)

→ 4 (Option 3) (Ans.)

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