6. Dividend, Divisor, Test of Divisibility,

 

DIVISIBILITY

Learn and Remember: 

(1) When one number can be completely divided by another number, then the first number is said to be divisible by the second.

(2) The digit in the units place of the product of two numbers is the same as the digit in the units place of the product of the units place digits of those two numbers. eg. 357 x 19=6783. The digits in the units place of the numbers are 7 and 9.7 x 9 = 63. Units place digit in both the products is 3.

(3) If a number is multiplied by itself several times, then the digits in the units place of the products recur after equal intervals.

The following table shows the digits in the units place of the products when a number is multiplied by itself several times.

(4) In the operation of division,

Dividend = (Divisor x Quotient) + Remainder.

5. Tests for Divisibility :

(1) Divisibility by 2, 4, 8:

(a) If 0, 2, 4, 6, 8 occur in the units place of a number, then that number is divisible by 2

e.g. 20, 76, 214, 978, ..., are divisible by 2.

(b) If the number formed by the last two digits (tens and units place) of a given number

Is divisible by 4, then the given number also is divisible by 4.

e.g, in the number 1236, 36 is divisible by 4. 1236 is divisible by 4.

(c) If the number formed by the last three digits (hundreds, tens and units place) of a

given number is divisible by 8, then the given number also is divisible by 8.

e.g. In the number 2856, 856 is divisible by 8.2856 is divisible by 8.

(2) Divisibility by 3, 6, 9, 18

 (a) If the sum of the digits of a number is divisible by 3, then that number also is divisible by 3.

  ∴ 8946 ÷ 3 =     →  8+9+4+6 = 27 ÷ 3 → Divisible by 3

(b) If the sum of the digits of an even number is divisible by 3, then that number is divisible by 6. ( 2 x 3 = 6 )

(c) If the sum of the digits of a given number is divisible by 9, then the given number also is divisible by 9.

∴ 8946 ÷ 9 =     →  8+9+4+6 = 27 ÷ 9 → Divisible by 9

(d) If the sum of the digits of an even number is divisible by 9, then that number is divisible by 18. ( 2 x 9 = 18 )

e.g. the sum of the digits of 5832 is 5 +8+ 3 + 2 = 18. 18 is divisible by 3 as well as 9.

∴ 5832 is divisible by 3 and 9.

Also, 5832 is an even number. So it is divisible by 6 and 18 as well.

(iii) Divisibility by 5, 10 :

(a) If the digit 0 or 5 occurs in the units place of a number, then that number is divisible by 5.

e.g. the numbers 35, 975, 1020, ..., etc. are divisible by 5.

(b) If 0 occurs in the units place of a number, then that number is divisible by 10.

e.g. 20, 470, 1090, ..., etc. are divisible by 10.

(iv) Divisibility by 7 :

If twice the digit at the units place of the given number is subtracted from the number formed by the remaining digits is zero or divisible by 7, then the number is divisible by 7. e.g.

(a) 126. Here, twice the digit at the units place is 6 x 2 = 12. The number formed by the remaining digits is 12. 12 – 12 = 0. :the number 126 is divisible by 7.

(b) 26502. Here, 2 x 2 = 4. 2650 – 4=2646. Again, 6 x 2 = 12 and 264 – 12= 252.

252 is divisible by 7. :. 26502 is divisible by 7. OR take one more step. 252. Here,

2 x 2 =4 and 25 – 4 = 21. 21 is divisible by 7.

∴ 26502 is divisible by 7.

(v) Divisibility by 11:

   If the sum of the digits in the odd places of a number equals the sum of the digits in its even places or if their difference is divisible by 11, then the given number also is divisible by 11. e.g.

(a) In the number 1243, the sum of the odd place digits = 1+4=5

the sum of the even place digits = 2+3=5. : 1243 is divisible by 11.

(b) In the number 11979, the sum of the odd place digits = 1 +9+9 = 19

the sum of the even place digits = 1 +7=8. Difference between them = 19–8=11.

This is divisible by 11. .. 11979 is divisible by 11.

Divisibility by 12	4 x 3
Divisibility by 15	3 x 5
Divisibility by 27	9 x 3