Problems based on the divisibility test

1. Replace the sign ___ by the smallest number to make them divisible by the given number.

a. 7158__ by 6

By the divisibility test,
if the number is divisible by 2 and 3 then that number is divisible by 6.
if the given number is divisible by 2 the units place must be 0 , 2, 4 , 6 , or 8.
if the given number is divisible by 3 then first we have to add the digits 7+1+5+8=21
21 is divisible by 3
so we can replace the sign by 0 since 71580(unit place 0) which is divisible by 2 and 71580(add will get 21) which is divisible by 3.
hence 71580 is divisible by 6

b. 8152__ by 4

By the divisibility test,
if the last two digits is divisible by 4 then that number is divisible by 4.
if we replace the sign by 0 then the last two digits be 20 which is divisible by 4.
hence 81520 is divisible by 4

c. 86__72 by 11

By the divisibility test,
the sum of the numbers in odd place = 8 + __+2  = 10+__
sum of the numbers in even place = 6 + 7  = 13
10+__  - 13 = 0 (since the difference must be either 0 or divisible by 11)
[we have to write the smallest possible number hence we choose 0]
so __ be 3
13-13=0
hence 86372 is divisible by 11

d. 631__24 by 8

By the divisibility test,
if the last three digit is divisible by 8 then that number is divisible by 8.
__24 , 24 itself is divisible by 8 hence we can replace the sign by 0
hence 631024 is divisible by 8.

2. State whether the following sentences are true or false.

a. 231648 is divisible by 4 and 8

Divisible by 4
lets consider last two digits 48 which is divisible by 4.
Divisible by 8
lets consider last three digits 648 which is divisible by 8
hence 231648 is divisible by 4 and 8 TRUE

b. 976575 is divisible by 3 and 15

Divisible by 3
lets add all the digits 9+7+6+5+7+5=39 which is divisible by 3
Divisible by 15
if the number is divisible by 3 and 5 then that number is divisible by 15
we already found that number number is divisible by 3
the unit place is 5 so the given number is divisible by 5 
hence it is divisible by 15
976575 is divisible by 3 and 15 TRUE


c. 567835 is divisible by 5 and 25  

Divisible by 5
the unit place in the given number 5, so the given number is divisible by 5
Divisible by 25
if the last two digits are 00,50 or 75 then that number is divisible by 25
here the last two digits is 35, so 567835 is not divisible by 25
567835 is divisible by 5 and 25 FALSE  

3. What digit you add each time to make the numbers on the Violet divisible by the numbers given in the RED

a. 67984311 by 2,3,4,5,6,8,9,10

Divisible by 2
if we add last digit as 0 then according to the divisibility test,
679843110 is divisible by 2

Divisible by 3
lets add all the digits 6+7+9+8+4+3+1+1+__ = 39+__
39 itself is divisible by 3
hence we can replace the sign as 0
679843110 is divisible by 3

Divisible by 4
the last two digits 1__, if we replace sign by 2 then 12 which is divisible by 4
hence 679843112 is divisible by 4

Divisible by 5
the last digit must be either 0 or 5
hence it can be 679843110 or 679843115 which are divisible by 5

Divisible by 6
According to the divisibility test,
the number must be divisible by both 2 and 3
if we add 0 that is 679843110 is divisible by both 2 and 3.
hence 679843110 is divisible by 6

Divisible by 8
lets consider the last three digits 11__ ,
110,111 are not divisible by 8
112 is divisible by 8
hence 679843112 is divisible by 8

Divisible by 9
Sum of the digits = 39 + __ 
39 is not divisible by 9
if we add 6 to 39 which gives 45 and 45 be divisible by 9
hence 679843116 is divisible by 9

Divisible by 10
the last digit must be 0
hence 679843110 is divisible by 10 

 

Tests for Divisibility

Divisibility by 2

If we observe a few multiples of 2 to be 2,4,6,8,10,12,14,...
that is the unit place of these numbers are like 0 , 2 , 4 , 6 , 8.
hence we can conclude that a number is divisible by 2 if that number has any of the digits
 0 , 2 , 4 , 6 ,or  8 in its units place.

Divisibility by 3

If the sum of the digits is a multiple of 3, then the number is divisible by 3.
Example
669 , here first we have to add all the digits
6+6+9 = 21
21 is divisible by 3
hence 669 is divisible by 3

Divisibility by 4

A number with 3 or more digits is divisible by 4 if the last two digits of the given number is divisible by 4.
Example
8216
lets consider the last two digits
16 which is divisible by 4
hence 8216 is divisible by 4

Divisibility by 5

A number which has either 0 or 5 in its units place then that number is divisible by 5.

Divisibility by 6

If the number is divisible by 2 and 3 both then that number is divisible by 6

Divisibility by 8

If the number formed by the last three digits is divisible by 8 then that number is divisible by 8.

Example
86512
lets consider the last three digits 512 which is divisible by 8.
hence 86512 is divisible by 8

Divisibility by 9

If the sum of the digits of the given number is divisible by 9 then that number is divisible by 9.

Example
504
lets add all the digits
5+0+4 = 9 which is divisible by 9
hence 504 is divisible by 9

Divisibility by 10

If the number has 0 in its units place then that number is divisible by 10

Divisibility by 11

First we have to find the difference between the sum of the digits at odd places (from the right side)
and the sum of the digits at even places of the number.If the difference is either 0 or that number is divisible by 11 then the given number is divisible by 11.
Example
61809
9+8+6 = 23
0+1=1
their difference = 23-1 = 22 which is divisible by 11
hence 61809 is divisible by 11

Math : Grade 5 : Exercise 2E

Solve. Use models to help you.

1. a. 612 + ______ = 948

Given,
What number if we add with 612 will give you 948
so, if we subtract 612 from 948 we will get the number
that is, 948 - 612 = 336

b. 7394 + ________ = 12642

12642 - 7394 = 5,248

c. ______ - 847 = 1238

Given,
what if we subtract 847 from what number will get 1238
so, if we add 847 with 1238 we will get that number
that is, 847 + 1238 = 2,085

d. ___________ - 9162 = 1811

9162 + 1811 = 10,973

e. 9408 - _______ = 1138

Given
If we subtract a number from 9408 will get 1138
so, if we subtract 1138 from 9408 will get that number
that is, 9408 - 1138 = 8,270

f. 49584 - _______ = 23175

49584 - 23175 = 26,409

2. a. What must add to 18345 to make it 19624?

Given
_______ + 18345 = 19624
that is, 19624 - 18345 = 1,279

b. If you subtract 23146 from a number , you are left with 35906. Find the number.

Given
__________ - 23146 = 35906
that is, 35906 + 23146 = 59,052

c. Find the number which must be subtracted from 83196 to leave 11422.

Given
83196 - _______ = 11422
that is, 83196 - 11422 = 71,774

d. The sum of two numbers is 40132. If one number is 29184, find the other number.

Given
29184 + _______ = 40132
that is, 40132 - 29184 = 10,948

3. a. Khalid wants to buy a board game that cost Rs 501. He has Rs 479. How much more money does he need?

Given
Cost of a board game = Rs 501
Amount he had = Rs 479
we have to find how much more money he need.
that is, 479 + _______ = 501
so, 501 - 479 = 22
Hence he need Rs22

b. Harshita's stamp album can hold 1,500 stamps. So far she has pasted 785 stamps in it. How many more stamps can she paste in it?

Given
Number of stamps album can hold = 1,500
Number of stamps she pasted = 785
we have to find how many more stamps she can paste
that is, 785 + ______ = 1,500
so, 1500 - 785 = 715
hence she can paste 715 more

c. A large library has lent out 1,785 books. It has 7,816 books left. How many books does the library have in all?

Given,
Number of books has lent out = 1,785
number of books left = 7,816
total number of books the library have = 1,785 + 7,816
                                                              = 9,601
hence the library have 9601 books

d. An art exhibition had 915 pieces of art on show. Some of them got sold but there were 211 unsold pieces. How many pieces were sold?

Given
Number of pieces of art on show = 915
number of pieces unsold = 211
we have to find number of sold out pieces.
number of pieces were sold = 915 - 211
                                             = 704
hence 704 pieces were sold

Math : Grade 5 : Exercise 2D

1 Solve using addition, subtraction, multiplication or division

a. The Sunshine club newspaper printed 33,530 copies in a year. Of these 28,395 copies were distributed. How many were not distributed?

Given,

Total copies printed in a year = 33,530

Number of copies distributed = 28,395

We have to find the number of copies which are not distributed

That is,

Number of copies not distributed = Total copies - Number of copies distributed

                            = 33,530 - 28,395

                            = 5,135

Hence 5,135 copies weren’t distributed.

b. The odometer on a van showed 53,811 km in the beginning of October. After being used for three months it showed 84,209 km in December. If it had traveled 21,614 km in October and November, how much distance did it cover in the month of December?

Given,

In the beginning of October the odometer showed = 53,811 km

In December it showed = 84,209 km

Therefore,

Distance traveled from October to December = 84,209 - 53,811

                                   = 30,398km

Also given,

Distance traveled in October and November = 21,614km

We have to find

The distance it cover in the month = total distance from October to December - distance covered in October and November

                            = 30,398 - 21,614

                            = 8,784 km

c. Sushil’s car traveled 25,384 km in one year and suraj’s car traveled 30,001 km in the same year. How many kilometer less did Sushil’s car run?

Given,

Distance traveled by Sushil’s car = 25,384 km

Distance traveled by Suraj’s car = 30,001 km

We have to find,

Number of Kilometer less did Sushil’s car run = Distance traveled by Suraj’s car - Distance traveled by Sushil’s car

                                     = 30,001 - 25,384

                                     = 4,617 km

d. Mr Shenoy had Rs3,25,765. He borrowed Rs1,12,700 to buy a new car. How much did the car cost?

Given,

Amount Mr Shenoy had to buy a new car = Rs 3,25,765

Amount he borrowed to buy a new car = Rs 1,12,700

Total amount of the car cost = 3,25,765 + 1,12,700

                       = Rs 4,38,465

e. Sriram won 75 tournaments. The prize money totals up to Rs 2,25,000. If he received the same amount for every tournament, how much had he earned per tournament?

Given,

Total number of tournaments = 75

Total Prize money = Rs 2,25,000

Amount he earned per tournament = total prize money / total number of tournaments

                              = 2,25,000 / 75

                              = Rs 3,000

f. A school needs 24,510 pencils a year. How many boxes of 25 must the school buy?

Total pencils school needs a year = 24,510

A box contains 25 pencils each

Hence total number of boxes school needs = 24,510 / 25

                                   = 980.4

Boxes should not be in decimal.

Hence 981 boxes so that no more pencils remains.

g. The toy store had 20 boxes of dolls and 25 boxes of teddy bears. Each box holds 24 of each. How many dolls and teddy bears did the toy store have in all?

Given,

Number of boxes of dolls = 20

Number of boxes of teddy = 25

Each box contains 24 of respective toys.

Total number of dolls = number of boxes of dolls * number of dolls per box

                  = 20 * 24

                  = 480 dolls

Total number of teddy = number of boxes of teddy * number of teddy per box

                   = 25 * 24

                   = 600 teddy bears

Hence total number of toys = 480 dolls + 600 teddy bears

                       = 1080 toys