If the 9-digit number 5x79856y6 is divisible by 36, then what is the negative value of √(2x + y) for the largest possible value of y, given x and y are natural numbers?

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Accepted Answer

Step 1: Understand divisibility by 36
A number is divisible by 36 if and only if it is divisible by both 4 and 9.

Step 2: Divisibility by 4
A number is divisible by 4 if its last two digits form a number divisible by 4.

The last two digits of the number are 'y6'. So, the number formed is 10*y + 6.

Check for divisibility by 4:
(10*y + 6) % 4 = 0

Since 10 % 4 = 2, the expression becomes:
(2*y + 6) % 4 = 0

Try values of y (natural numbers, i.e., 1, 2, 3, ...):
- y=1: 2*1+6=8 %4=0 ✓
- y=2: 2*2+6=10 %4=2 ✗
- y=3: 2*3+6=12 %4=0 ✓
- y=4: 2*4+6=14 %4=2 ✗
- y=5: 2*5+6=16 %4=0 ✓
- y=6: 2*6+6=18 %4=2 ✗
- y=7: 2*7+6=20 %4=0 ✓
- y=8: 2*8+6=22 %4=2 ✗
- y=9: 2*9+6=24 %4=0 ✓

Possible y values for divisibility by 4 are: 1, 3, 5, 7, 9.

Step 3: Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

Sum of digits = 5 + x + 7 + 9 + 8 + 5 + 6 + y + 6
Calculate sum excluding x and y:
5 + 7 + 9 + 8 + 5 + 6 + 6 = 46

So, total sum = 46 + x + y

For divisibility by 9:
(46 + x + y) % 9 = 0

Step 4: Find x and y
We want the largest possible y from {1,3,5,7,9}.
Try y=9:
(46 + x + 9) % 9 = (55 + x) % 9 = 0

55 % 9 = 1 (since 9*6=54)
So, (1 + x) % 9 = 0 => x % 9 = 8

Since x is a natural number (1 to 9), x=8.

Check if x=8 and y=9 satisfy all conditions:
- Divisible by 4: y=9 ✓
- Divisible by 9: (46 + 8 + 9) = 63, 63 % 9=0 ✓

Step 5: Calculate √(2x + y)
2x + y = 2*8 + 9 = 16 + 9 = 25

√25 = 5

Step 6: Negative value of √(2x + y) is -5.

Hence, the answer is -5.

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