Numbers and Divisibility

All of the above will divide 2 because the units digit is even.

The first choice is 2. The number so obtained is 16352. Now, the sum of the digits = 1 + 6 + 3 + 5 + 2 = 17 As 17 is not divisible by 3, so the number so formed will not be divisible by 3. The second choice is 4. The number so obtained is 16354. Now, the sum of the digits = 1+ 6 + 3 + 5 + 4 = 19 As 19 is not divisible by 3, so the number so formed will not be divisible by 3. The third choice is 0. The number so obtained is 16350. Sum of digits = 1 + 6 + 3 + 5 + 0 = 15. Now 15 is divisible by 3, therefore the number will also be divisible by 3. The fourth choice is 5. The number so obtained is 16355. Now, the sum of the digits = 1 + 6 + 3 + 5 + 5 = 20 As 20 is not divisible by 3, so the number so formed will not be divisible by 3.  

A number divisible by 2, 3 and 5 is of the type 2 x 3 x 5 x a = 30 a. Out of the given options, only 30 is that type of a number. Hence, option (5) is correct.

A number is divisible by 3 if the sum of the digits of the number is divisible by 3. 

The smallest number that is exactly divisible by18,20 and 24.we take their lowest common mutipleL.C.M. of 18, 20, 24 is 

2 x 2 x 2 x 3 x 3 x 5 =360Number just greater than1500 is divisible by 360 1500 / 360=Quotient = 4 Remainder = 60 Number which will be added 1500+ ( 360 - 60 ) 1500 + 300 = 1800 Required number is 300 when added to1500 will be exactly divisible by18,20,24 As we division is repeated subtraction (1)  1500 - 360 = 1140(2)  1140 - 360 = 780(3) 780 - 360 = 420(4) 420 - 360 = 60

To make last 5th set we will add 300 to 60 to bring 0 60 + 300 = 360(5) 360 - 360  = 0 

so number 300 which will be added to 1500

 231414 is not divisible by 4.

A number is divisible by 8 if the last 3 digits of the number are evenly divisible by 8.

 m = 2 is one of the possible values of m.

If the number formed by the last two digits of a number is divisible by 4, then the number is also divisible by 4. Here, the number is 13. Also, 13 is not divisible by 4. Therefore, 256413 is not divisible by 4.

For a number to be divisible by 9, the sum of the digits of the number should be divisible by 9. For 263125, the sum of the digits = 19. As we know 19 is not divisible by 9, so 231462 is also not divisible by 9.

If m = 2, then K becomes 52362. Last two digits of K = 62 and 62 is not divisible by 4. Therefore, 52362 is also not divisible by 4.

For a number to be divisible by 6, the number must be an even number and also the number should be divisible by 3. Here, 521424 is an even number, also the sum of the digits is 18, which is divisible by 3 and hence, the number is also divisible by 3. Therefore, the given number is divisible by 6.

If the sum of all the digits is divisible by 3, then the number is also divisible by 3. Therefore, sum of digits = 2 + 1 + 4 + 5 + 1 + 1 = 1414 is not divisible by 3. Therefore, 214511 is also not divisible by 3

For a number to be divisible by 8, the last three digits of the number must be divisible by 8. For 216664, last three digits come out to be 664 which is divisible by 8. Hence, the number is also divisible by 8.

If k is divisible by 3, then the sum of the digits must be divisible by 3. Here, k = 231mn05. Therefore, 2 + 3 + 1 + (m + n) + 0 + 5 = L (say). Putting the value of (m + n) = 1, we get L = 16, which is not divisible by 3. Hence, for (m + n) = 5, k will not be divisible by 3.