### 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:

- 4
- 6
- 5
- 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.

What are co-primes? Numbers that do not have any common factors. For example, 7 and 8 are co-primes but 6 and 8 are not co-primes because the latter have 2 as a common factor.985x3678y to be div by both 8 and 9

NCERT class VI Math: A number which is a product only of co-primes, is divisible by each of them.[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

If you know only the divisibility criterion for 8, and nothing else, then this method helps.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

If you know only the divisibility criterion for 9, and nothing else, then this method helps.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).

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