How to find numbers of numbers divisible by a certain numbers

It will be explained through an example.

Q1. How many number from 1 to 100 are divided by 6?

Explaination:

We will divide 100 by 6 and then write the divided and remainder thus obtained.

100 = 16 x 6 + 4

Thus there are 16 numbers that are divided by 6.

This can be done for other numbers also.

Q2. How many numbers between 200 and 500 are divisible by 13.

Explanation:
For such questions, 200 and 500 are excluded from the answer even if they are divisible by the given number.
Now , we will divide both 200 and 500 by 13 and write their dividend and remainders.

200 = 15 x 13 + 5

adding 8 to the above equation on both the sides to get
208 = 16x13

Now, dividing 500 by 13 to get
500 = 38 x 13 + 6
Subtracting 6 from both the sides to get
494 = 38x13

Thus between 200 and 500 there is 16th multiple of 13 to 38th multiple. To get the required answer we subtract 16 from 38 and add 1 to the answer (as both the numbers are incuded.)

Thus there are 38 - 16 + 1 = 23 numbers between 200 and 500 that are divisible by 13.

Quicker method :  There are 500-200 = 300 number between 200 and 500. Now divide 300 by 13 to get
300 = 23 x 13 + 1
Thus there are 23 number between 200 and 500 that are divisible by 13.

Q3.What is the least number which must be subtracted from 9600 so that the remaining number becomes divisible by 78?
Explanation;
Divide the number 9600 by 78 to get
9600 = 123 x 78 +  6
Thus we have to subtract 6 from 9600 to get a number divisible by 78 which is 9600 -6 = 9594

Q3. Find the sum of all numbers between 200 and 600 which are divisible by 16.

Explanation:
First of all we have to find the first number just after 200 which is divisble by 16 and a number that is just less than 600 which is divisible by 16.

Divide 200 by 16 to get
200 = 16x12 + 8
We have to add 8 to 200 to get a number just greater than 200 which is divisible by 16.
That number is 200 + 8 = 208
Now for the second number again divide 600 by 16, to get
600 = 37x16 +8
Thus we have to subtract 8 from 600 to get a number divisible by 16.
which is 600 - 8 = 592
And number of such numbers between 200-600 are 37-13 + 1 = 25
After getting the first and the last number divisible by 16 in the range 200 - 600 and the number of such numbers, we have to use the formula for A.P.

Required sum = (25)(208+592)/2 = 25 .400 = 10000

Q4. What least value
 should be given to * so that the number 91876*2 is divisible by 8?
Explanation:
For a number to be divisible by 8 , last three digits of the number should be divided by 8.
Last three digits of 91876*2 are 6*2.
Now checking for various digits starting from 0.
If * is 0 then the number is 602 which is not divisible by 8.
If * is 1 then the number is 612 which is not divisible by 8.
If * is 2 then the number is 622 which is not divisible by 8.
If * is 3 then the number is 632 which is divisible by 8.
Hence the least value for * is 3.