Question

If 653xy is exactly divisible by 80, then find the value of (x + y).
        (A). 2
        (B). 3
        (C). 4
        (D). 6

Answer 1

Hint: The number is given in the form of 3 digits and 2 variables. Divide the given number (80) into the product of 2 numbers. Now, look at the divisibility rules of both numbers. The number with variables must be divisible by these both numbers for it, being divisible by their product which is 80. By these rules, you can get the value of x, y in 2 cases. As there are 2 rules. After getting the values of x, y just add them together to get the value of their sum. Which is asked in the question. So, their sum (x + y) will be your result.

Complete step-by-step solution -
Divisibility Rule of 8: For a number to be divisible by 8, it’s last three digits must be a multiple of 8.
Divisibility Rule of 10: For a number to be divisible by 10, the numbers must have units digit as 0.
The condition required is that the given number must be divisible by 80.
We can write 80 as a product by 8, 10. 80 = (8) (10).
So, if a number is divisible by 8, 10 it will be divisible by 80.
Now we need to apply the divisibility rule by 8 and divisibility rule of 10 to prove that number is divisible by 80.
Given number in the question, can be written as follows: -
653xy
For 653xy to be divisible by 10, the unit digits must be 0.
By applying the above condition, we get the value of y to be:
y = 0 – (1)
For 653x to be divisible by 8, the number 53x must be divisible by 8.
By applying the above condition, we get the value of x as;
53x must be divisible by 8.
53x can be written as = \[5\times 100+3\times 10+x\]
As, \[500=62\times 8+4\]; \[30=8\times 3+6\]
By substituting these into above, we get
\[\left( 62\times 8+4 \right)+\left( 8\times 3+6 \right)+x=\left( 65\times 8 \right)+\left( 10+x \right)\]
By this, we get (10 + x) must be divisible by 8 where possible is x = 6. Because x is a digit, where \[0\le x\le 9\].
So, x = 6, y = 0 that implies x + y = 6.
Therefore, option (d) is correct.

Note: Don’t forget to remove y after checking the divisibility rule of 10. Because 10 will cancel the y, when 8 is remaining for the division the number will become 653x. Generally, students consider that y and end up getting x = 2. But it is wrong you must remove y after getting it as 0, then you will get the right answer.