Numbers and Divisibility

 (k + m) cannot be 0.

Required number = LCM + the number subtracted

= LCM of 30, 32 and 42 + 9

= 3360 + 9

= 3369

So, the required number is 3369.

Let the greatest 4-digit number be 9999.

Now, on dividing 9999 by 78,

Dividend = Quotient x Divisor + Remainder

9999 = 12 x 78 + 15

So, the required number is 9999 - 15 = 9984.

Required number = LCM of 23, 29 and 31 + 5000

= 25677

The greatest 4-digit number is 9999.

Now, LCM of 12, 27 and 9 = 108

So, on dividing 9999 by 108, 

Dividend = quotient x divisor + remainder

9999 = 92 x 108 + 63

Therefore, required number = 9999 - 63 = 9936

Among three consecutive numbers, there must exist atleast one even number. So, the product of three consecutive numbers will always be divisible by 2.

Sum of the digits of 152658 = 1 + 5 + 2 + 6 + 5 + 8 = 9 9 is divisible by 9. Hence, 152658 is divisible by 9.

Numbers divisible by both 6 and 10 are 30, 60,90,120, .... All these are not divisible by 12.

LCM of 3, 4 ,5 and 6, which is equal to 60. (You can also argue that 30 is not divisible by 4, 40 is not divisible by 3, and 60 is divisible by all the given numbers and must be the answer.)

As GCD or HCF is 13, so the pair must be divisible by 13.

Zero cannot be used as a divisor because division by zero is undefined in mathematics. When you divide a number by zero, the result has no meaningful value. This is a fundamental rule in arithmetic.

Sum of digits of 32541679 = 37, which is not divisible by 9. Hence, the number is also not divisible by 9. Sum of digits of 32564872 = 37, which is not divisible by 9. Hence, the number is also not divisible by 9. Sum of digits of 32587416 = 36, which is divisible by 9. If the sum of the digits is divisible by 9, then the number itself is divisible by 9. Hence, 32587416 is divisible by 9. Sum of digits of 32546874 = 39, which is not divisible by 9. Hence, the number is also not divisible by 9.  

Out of the given options, only 90 is divisible by both 10 and 18.