(solved)Question 7 SSC-CGL 2018 June 4 Shift 1

Question 7
SSC-CGL 2018 June 4 Shift 1

If a number of 9 digits is 985x3678y, the number is divisible by 72, then the value of (4x - 3y) will be:

1. 4
2. 6
3. 5
4. 3

Solution in Short

985x3678y is divisible by 72, which is the product of 8 and 9, which are two co-primes. So it should be divisible both by 8 and 9.

985x3678y is divisible by 8 if 78y is divisble by 8, which gives y = 4. Our number becomes 985x36784, which will be divisible by 9 if the sum of the digits, 50 + x is divisible by 9, which gives y = 4.

Finally, (4x - 3y) = 4 is the answer!

Video Explanation (2 minutes)

Please watch this video for a good explanation of the above solution:

Solution in Detail

985x3678y is divisible by (8 x 9)

But 8 and 9 are co-primes.

985x3678y to be div by both 8 and 9

[1] First take the divisibility by 8

The number formed by last 3 digits, i.e., $\displaystyle \underline{\text{78y}}$ should be divisible by 8.

By hit and trial the number is $\displaystyle 78\underline{4} $

$\displaystyle \implies y = 4 \text{ . . . (1)}$

The number so far: 985x36784

[2] Next take the divisibility by 9

sum of digits is to be divisible by 9.

sum of digits of 985x36784 = 50 + x

$\displaystyle \therefore (50 + x)$ to be divisible by 9.

$\displaystyle 54 \equiv (50 + 4)$ is in the table of 9

Which gives $\displaystyle x = 4$

$\displaystyle \therefore (4x - 3y) = 4\cdot 4 - 3\cdot 4 $

$\displaystyle = 4 \:\underline{Ans}$

Solution by Alternate Method

By the divisibility criterion for 8, the last three digits should be divisible by 8. So 78y should be divisible by 8. Since, 784/8 = 98, this means y should be 4.

Now 4x - 3y will be $\displaystyle 4x - 3 \times 4 = 4x - 12$. Cycle through the options.

option (a) 4x - 12 = 4 gives x = 4, could be possible.

option (b) 4x - 12 = 6 gives x = 9/2, impossible.

option (c) 4x - 12 = 5 gives x = 17/4, impossible.

option (d) 4x - 12 = 3 gives x = 15/4, impossible.

Hence the answer is 4x - 3y = 4, option (a).

Solution by Another Alternate Trick

We have to find (4x - 3y). Or, we can say that we have to find x + 3(x - y).

Keeping the options in mind, we can conclude that (x - y) cannot be more than 1 [why? because if (x - y) were 2, say, then x + 3 (x - y) would be more than 6, and none of the options is greater than 6]

So x and y are either equal or consecutive.

By the divisibility rule of 9, the sum of digits = 46 + x + y should be divisible by 9. Hence x + y = 8

TIP: Don't do full addition, neglect 9 and all those pairs that add to 9 because they are already multiples of 9. $\displaystyle \cancel{9}85\text{x}\cancel{36}78\text{y}$ = (28 + x + y) gives x + y =8 again.

The only possibilities is x = y = 4. [Why? because out of (1,7), (2, 6), (3, 5) and (4, 4), only (4, 4) meets the criterion of (x, y) being either consecutive or equal!]

Hence the answer is 4x - 3y = 4, option (a).