Divisibility by 5 and 10:

Statement : An integer a is divisible by 5 if and only if it ends with 5 or 0. A number is divisible by 10 if and only if it ends with 0. 

---

Example: How many numbers from 101 to 200 are divisible by 5?

Solution:

Write the numbers from 101 to 200 which end with 5 and 0. i.e.,

105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200. There are 20 such numbers.

---

Example: Is the number 12345 is divisible by 15?

Solution:

Note that 15 = 3 x 5. Here again the given number 12345 must be divisible by both 3 and 5.

The sum of the digits is 1+2+3+4+5 = 15, which is divisible by 3 and the given number 12345 ends with 5, so it’s also divisible by 5. Therefore, 12345 is divisible by 15.

---

Example : How many numbers from 201 to 250 are divisible by 5, but not by 3?

Solution:

The numbers which are divisible by 5 from 201 to 250 are 205, 210, 215, 220, 225, 230, 235, 240, 245, 250. Now let us compute the digital sum of these numbers:

205 ——-2+0+5 = 7, not divisible by 3

210 ——– 2+1 = 3, divisible by 3

215 ———2+1+5 = 8, not divisible by 3

220———–2+2+0 = 4, not divisible by 3

225———-2+2+5 = 9, divisible by 3

230———-2+3+0 = 5, not divisible by 3

235———-2+3+5 = 10, not divisible by 3

240———2+4+0 = 6, divisible by 3

245———-2+4+5 = 11, not divisible by 3

250———–2+5+0 = 7, divisible by 3

---