Divisible by 6 - Try with a Game!

In this post, you would learn about numbers that are divisible by 6 through a game which is a numerical maze!

First keep in your mind that:

If the sum of the digits of a number is divisible by 3 AND its last digit is even then that number is divisible by 6. For example, 912 is divisible by 6 because the sum of its digits, twelve (9 + 1 + 2 = 12), is divisible by 3 and its last digit, two, is an even number.

Now, find a path from the START to the END passing only through numbers that are divisible by 6. Note that there is only one solution for this numerical maze.

Scroll down to check your solution!

Solution:

Divisible by 12 - Try with a Game!

In this post, you would learn about numbers that are divisible by 12 through a game which is a numerical maze!

First keep in your mind that:

* If a number is divisible by 3 and by 4 then the number is divisible by 12.

* If the sum of the digits of a number is divisible by 3 then the number is divisible by 3. For example, 2130021 is divisible by 3 because 9, the sum of its digits (2 + 1 + 3 + 0 + 0 + 2 + 1 = 9), is divisible by 3.

* If half of a number is an even number then that number is divisible by 4. For example, 72 is divisible by 4 because half of it, 36, is an even number. Also, if the last two digits of a number is divisible by 4 then that number is divisible by 4. As an example: 897232 is divisible by 4 because its last 2 digits; i.e. 32, is divisible by 4.

Now, find a path from the START to the END passing only through numbers that are divisible by 12. Note that there is only one solution for this numerical maze.

Scroll down to check your solution!

Solution:

Square of 101, 102, 103, ..., 1001, 1002, ..., ...

Consider the following squares:

101 * 101 = 10201
102 * 102 = 10404
103 * 103 = 10609
104 * 104 = 10816
...
109 * 109 = 11881

In all of the above squares, there is a simple pattern as described in the following.

101 * 101 = <-- five digits
10201
10201 <-- this bold '1' is calculated as 1*1 (101 * 101)
10201 <-- this bold '2' is calculated as 1+1 (101 * 101)
10201 <-- this bold '1' is due to 100*100=10000 (five digits)

Let's see for 102:

102 * 102 = <-- five digits
10404
10404 <-- this bold '4' is calculated as 2*2 (102 * 102)
10404 <-- this bold '4' is calculated as 2+2 (102 * 102)
10201 <-- again, this bold '1' is due to 100*100=10000 (five digits)

Now, let's see for 103:

103 * 103 = <-- five digits
10609
10609 <-- this bold '9' is calculated as 3*3 (103 * 103)
10609 <-- this bold '6' is calculated as 3+3 (103 * 103)
10609 <-- again, this bold '1' is due to 100*100=10000 (five digits)

Now, for 109:

109 * 109 = <-- five digits
11881
11881 <-- this bold '81' is calculated as 9*9 (109 * 109)
11881 <-- this bold '18' is calculated as 9+9 (109 * 109)
11881 <-- again, this bold '1' is due to 100*100=10000 (five digits)

This pattern can also be seen in 110, 111, ...

Now, consider the following squares:

1001 * 1001 = 1002001
1002 * 1002 = 1004004
1003 * 1003 = 1006009
1004 * 1004 = 1008016
...
1009 * 1009 = 1018081

It must be easy for you to extract the pattern!

Now try:

10000001 * 10000001
10000002 * 10000002
10000009 * 10000009
100000001 * 100000001
100000002 * 100000002
100000009 * 100000009

Squaring Two-Digit Numbers

To square 2-digit numbers follow the following algorithm.

Select an arbitrary number, for example 48 to follow:

1. Look for the nearest 10 (ceiling) boundary; for 48 it would be 50.

2. Take d = 50 - 48; so d = 2.

3. Take A = 50 * (50 - 2 * d) = 50 * 46 = 2300.

4. The answer is equal to A + d^2 = 2300 + 2 * 2 = 2304.

48 * 48 = 2304

Another example:

Select an arbitrary number to mentally calculate its square, for example 87.

1. Look for the nearest 10 boundary; for 87 it would be 90.

2. Take d = 90 - 87; so d = 3.

3. Take A = 90 * (90 - 2 * d) = 90 * 84 = 7560.

4. The answer is equal to A + d^2 = 7560 + 3 * 3 = 7569.

87 * 87 = 7569

Now let's try it for 73:

1. Look for the nearest 10 boundary; for 73 it would be 80.

2. Take d = 80 - 73; so d = 7.

3. Take A = 80 * (80 - 2 * 7) = 80 * 66 = 5280.

4. The answer is equal to A + d^2 = 5280 + 7 * 7 = 5329.

73 * 73 = 5329

Multiplying two numbers that differ from a number by the same small number

Follow the following examples to understand the simple idea behind it.

Example 1:

99 * 101 = (100 - 1) * (100 + 1) = (100 * 100) - 1 = 10000 - 1 = 9999

Rem:

99 * 101 = (100 * 100) - (1 * 1)

99 * 101 = 100^2 - 1^2

Example 2:

997 * 1003 = (1000 - 3) * (1000 + 3) = (1000 * 1000) - (3 * 3) = 1000000 - 9 = 999991

Rem:
997 * 1003 = (1000 * 1000) - (3 * 3)

999 * 1003 = 1000^2 - 3^2

Example 3:

47 * 53 = (50 - 3) * (50 + 3) = 2500 - 9 = 2491

Rem:
47 * 53 = (50 * 50) - (3 * 3)

47 * 53 = 50^2 - 3^2 

Now, do the following exercises:

1. 195 * 205 (Rem: 195 * 205 = 200^2 - 5^2)

2. 36 * 44

3. 77 * 83

4. 148 * 152

Answers:

1. 39975

2. 1584

3. 6391

4. 22496

Squaring Numbers Ending in Nine

To square a number ending in nine; e.g. 29, 59, 129, ..., first square the next number that is ending in zero, and then subtract both the number and the next number from it.


Example 1:

To square 29 (29^2):

29: the number
30: the next number

29^2 = 30^2 - (29 + 30) = 900 - 59 = 841


Example 2:

To square 69 (69^2):

69: the number
70: the next number

69^2 = 70^2 - (69 + 70) = 4900 - 139 = 4761


Example 3:

To square 129 (129^2):

129: the number
130: the next number

129^2 = 130^2 - (129 + 130) = 16900 - 259 = 16641

Squaring Numbers Ending in One

To square a number ending in one; e.g. 21, 51, 121, ..., first square the prior number that is ending in zero, and then add both the prior number and the number to it.


Example 1:

To square 21 (21^2):

21: the number
20: the prior number

21^2 = 20^2 + 20 + 21 = 441


Example 2:

To square 61 (61^2):

61: the number
60: the prior number

61^2 = 60^2 + 60 + 61 = 3721


Example 3:

To square 121 (121^2):

121: the number
120: the  prior number

121^2 = 120^2 + 120 + 121 = 14641

Squaring 14, 24, 34, ..., 94

To square a two digit number ending in 4; i.e. 14, 24, 34, ..., 94, first square the next number that is ending in 5 (as explained in "squaring a two digit number ending in 5"), and then subtract the number and the next number from it.


Example 1:

To square 14 (14^2):

14: the number
15: the next number
Based on "squaring a two digit number ending in 5", we know that 15^2 = 225. So:

14^2 = 225 - 14 - 15 = 196


Example 2:

To square 44 (44^2):


44: the number
45: the next number
Based on "squaring a two digit number ending in 5", we know that 45^2 = 2025. So:

44^2 = 2025 - 44 - 45 = 1936


Example 3:

To square 94 (94^2):


94: the number
95: the next number
Based on "squaring a two digit number ending in 5", we know that 95^2 = 9025. So:

94^2 = 9025 - 94 - 95 = 8836.

Squaring 16, 26, 36, ..., 96

To square a two digit number ending in 6; i.e. 16, 26, 36, ..., 96, first square the prior number that is ending in 5 (as explained in the previous post), and then add the number and the prior number to it.


Example 1:

To square 16 (16^2):

16: the number
15: the prior number
Based on the previous post, we know that 15^2 = 225. So:

16^2 = 225 + 16 + 15 = 256


Example 2:

To square 46 (46^2):


46: the number
45: the prior number
Based on the previous post, we know that 45^2 = 2025. So:

46^2 = 2025 + 46 + 45 = 2116



Example 3:


To square 96 (96^2):


96: the number
95: the prior number
Based on the previous post, we know that 95^2 = 9025. So:

96^2 = 9025 + 96 + 95 = 9216.

Squaring 15, 25, 35, ..., 95

To square a two digit number ending in 5; i.e. 15, 25, 35, ..., 95, first multiply the tens digit by the next whole number and then affix 25 to it.

Example 1:

To square 15 (15^2):

The tens digit of 15 is 1 and the next whole number of 1 is 2. So:
1 * 2 = 2
Now, we affix 25 to 2. The answer is 225.


Example 2:

To square 35 (35^2):

The tens digit of 35 is 3 and the next whole number of 3 is 4. So:
3 * 4 = 12
Now, we affix 25 to 12. The answer is 1225.


Example 3:

To square 95 (95^2):

The tens digit of 95 is 9 and the next whole number of 9 is 10. So:
9 * 10 = 90
Now, we affix 25 to 90. The answer is 9025.

Multiplying by 20, 25, and 30

Multiplying By Twenty

Multiplying by 20 is done in two stages, by 2 and then by 10.


Multiplying By Twenty Five

To multiplying by 25:

Divide the number by 4. Then:
If you have a remainder of 0 then add 00 at the end.
If you have a remainder of 1 then add 25 at the end.
If you have a remainder of 2 then add 50 at the end.
If you have a remainder of 3 then add 75 at the end.

Example: 143 * 25

143 / 4 = 35 remainder: 3

so
143 * 25 = 35 & 75 (for a remainder of 3 add 75 at the end)
143 * 25 = 3575


Multiplying By Thirty

Multiplying by 30 is done in two stages, by 3 and then by 10.

Multiplying by 5, 10, and 15

Multiplying By Five

Multiplying by 5 is just multiplying by 10 and then dividing by 2. Example:

15 * 5 = (15 * 10) / 2 = 150 / 2 = 75


Multiplying By Ten

Everybody knows that to multiply by 10, a zero should be added to the end of the number. Example:

26 * 10 = 260


Multiplying By Fifteen

To multiply by 15, you can use one of these two methods:

1) Multiply by 5 then then add it up with the product of the number and 10. Example:

43 * 15 = (43 * 5) + (43 * 10) = (43 * 10 / 2) + 430 = 215 + 430 = 645

In other words, add half of the product of the number and 10 to the product of the number and 10; i.e.:

43 * 15 = 430 + (1/2 * 430) = 645

2) Multiply by 30 and then divide by 2 (in the next post, a method is presented for "Multiply by 30"). Example:

57 * 15 = (57 * 30) / 2 = 1710 / 2 = 855

Multiply 11*11 to 19*19

Use this algorithm in your head:

Take 16 * 18 for an example:

1) Add 16 + 8 = 24
2) Multiply it by 10: 24 * 10 = 240
3) Add up the product of 6 and 8: 240 + (6 * 8) = 288
16 * 18 = 288

Another example:

14 * 17 = ?

1) Add 14 + 7 = 21
2) Multiply it by 10: 21 * 10 = 210
3) Add up the product of 4 and 7: 210 + (4 * 7) = 238
14 * 17 = 238

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Multiplying by 9, 99, 999, ...

Multiplying by 9 is as multiplying by 10-1. So, 9 * 8 is as (10 - 1) * 8 that is:

9 * 8 = (10 - 1) * 8 = 80 - 8 = 72

Another example:

26 * 9 = 26 * (10 - 1) = 260 - 26 = 234

To multiply by 99, one must multiply by 100-1:

53 * 99 = 53 * (100 - 1) = 5300 - 53 = 5200 + 100 - 53 = 5200 + 47 = 5247

Similarly, to multiply by 999, one must multiply by 1000-1:

87 * 999 = 87 * (1000 - 1) = 87000 - 87 = 86000 + 1000 - 87 = 86000 + 913 = 86913

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Numbers to Symbols: Strategies for Learning and Teaching Mathematics
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The Magic of 1001

Attach a 3-digit number to itself (e.g. 547 -> 547547). Divide it by 7.
The remainder will be zero!

Example: Take 826. Then 826826 / 7 = 118118

Now, divide the quotient by 11. The remainder will be zero!

Example:  826826 / 7 =  118118   -->   118118 / 11 = 10738

Again, divide the recent quotient by 13. The remainder will be zero!

Example:  826826 / 7 =  118118    -->   118118 / 11 = 10738  -->   10738 / 13 = 826

Where 826 is the selected number!

Another example:

Take 743!
Concatenate it with itself: 743743
Divide it by 7: 743743 / 7 = 106249
Divide the quotient by 11: 106249 / 11 = 9659
Divide the recent quotient by 13: 9659 / 13 = 743

Why?

7 * 11 * 13 = 1001
Multiplying 1001 by any 3-digit number will be as concatenating it with itself:

100 * 1001 = 100100

945 * 1001 = 945945

So, any number that is obtained by concatenating a 3-digit number with itself can be divided by 1001 and as 1001 is equal to 7 * 11 * 13, it can be divided by 7, 11, and 13.

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Rapid Math Tricks & Tips: 30 Days to Number Power
Numbers to Symbols: Strategies for Learning and Teaching Mathematics
50 Activities for Interpersonal Skills Training