Arithmetic1

LOGIC BEHIND MAGIC SQUARE: In a magic square 3x3, if we take numbers from 1 to 9 then the magic sum equal to 15. The magic square is as follows: 4 9 2  3 5 7 8 1 6 In the above magic square 3x3 magic sum is 15. In the above magic square 3x3 we are used numbers from1 to 9 arranged in 9 boxes. In any direction the sum equal to 15. Row wise  4+9+2=15, 3+5+7=15, 8+1+6=15. Column wise 4+3+8=15, 9+5+1=15, 2+7+6=15. Diagnol wise: 4+5+6=15, 2+5+8=15. The above magic square also can be written as follows: 5-1 5+4 5-3  5-2 5 5+2 5+3 5-4 5+1 In the above figure , shows the relation between middle number and other numbers. In the magic square we are taken numbers from 1 to 9 and number 5 is between 1 to 9. If we write  x in place of 5 in above magic square.                                         X-1 X+4 X-3  X-2 X X+2 X+3 X-4 X+1 Now we will write different magic square’s with different magic sum’s 2.MAGI

 MAGIC SQUARE: In a magic square 3x3, if we take numbers from 1 to 9 then the magic sum equal to 15. The magic square is as follows: 4 9 2  3 5 7 8 1 6 In the above magic square 3x3 magic sum is 15. In the above magic square 3x3 we are used numbers from1 to 9 arranged in 9 boxes. In any direction the sum equal to 15. Row wise  4+9+2=15, 3+5+7=15, 8+1+6=15. Column wise 4+3+8=15, 9+5+1=15, 2+7+6=15. Diagnol wise: 4+5+6=15, 2+5+8=15. The above magic square also can be written as follows: 5-1 5+4 5-3  5-2 5 5+2 5+3 5-4 5+1 In the above figure , shows the relation between middle number and other numbers. In the magic square we are taken numbers from 1 to 9 and number 5 is between 1 to 9. If we write  x in place of 5 in above magic square.                                         X-1 X+4 X-3  X-2 X X+2 X+3 X-4 X+1 Now we will write different magic square’s with different magic sum’s 2.MAGIC SUM=18. NU

To find out average of the prime numbers between 30 and 50 we need to find out the two things. Those are 1.      No Prime numbers between 30 and 50. 2.      Sum of prime numbers between 30 and 50. Prime numbers between 30 and 50 : 31,37,41,43,47 since there are 5 prime numbers are there between 30 and 50. Sum of prime numbers between 30 and 50: 31+37+41+43+47                                                                              =199 average of the prime numbers between 30 and 50:                     = sum of elements / no of elements                    =  199/5                    =  39.8 Since the average of prime numbers from 30 and 50 is 39.8

By seeing the prime number table we know that 25 prime numbers are existing in between 1 to 100 the following table explains Prime numbers 1-25 2,3,5,7,11,13,17,19,23 9 26-50 29,31,37,41,43,47 6 51-75 53,59,71,73,61,67 6 75-100 79,83,89,97 4 Total no of prime numbers 25 In the above table shows that from 1 to 25 numbers there are 9 prime numbers. similarly from 25 to 50 there are 6 prime numbers are there.like wise from 51-75 there are 6 prime numbers and from 75 to 100 there are 4 prime numbers there. The following table shows the sum of prime numbers Prime numbers sum 1-25 2+3+5+7+11+13+17+19+23 100 26-50 29+31+37+41+43+47 228 51-75 53+59+71+73+61+67 384 75-100 79+83+89+97 348 Sum of the prime numbers 1060 so sum of the prime numbers from 1 to 100 is 1060.

by seeing the prime number table we know that 25 prime numbers are existing in between 1 to 100 the following table explains Prime numbers 1-25 2,3,5,7,11,13,17,19,23 9 26-50 29,31,37,41,43,47 6 51-75 53,59,71,73,61,67 6 75-100 79,83,89,97 4   in the above table shows that from 1 to 25 numbers there are 9 prime numbers similarly from 25 to 50 there are 6 prime numbers are there. like wise from 51-75 there are 6 prime numbers and from 75 to 100 there are 4 prime numbers there. by observing the above tables the the following questions can ask. 1 .  What are the prime numbers between 20 and 50?    Ans:prime numbers : 23,29.31,37,41,43,47           since there are 7 prime numbers are there between 20 and 50. 2.  How many prime numbers are between 30 and 50?      Ans:prime numbers: 31,37,41,43,47            since there are 5 prime numbers are there between 30 and 50. 3. What are the prime numbers between 70 and 100

by seeing the prime number table we know that 25 prime numbers are existing in between 1 to 100 the following table explains Prime numbers 1-25 2,3,5,7,11,13,17,19,23 9 26-50 29,31,37,41,43,47 6 51-75 53,59,71,73,61,67 6 75-100 79,83,89,97 4   in the above table shows that from 1 to 25 numbers there are 9 prime numbers similarly from 25 to 50 there are 6 prime numbers are there. like wise from 51-75 there are 6 prime numbers and from 75 to 100 there are 4 prime numbers there. by observing the above tables the the following questions can ask. 1 .  What are the prime numbers between 20 and 50?    Ans:prime numbers : 23,29.31,37,41,43,47           since there are 7 prime numbers are there between 20 and 50. 2.  How many prime numbers are between 30 and 50?     Ans:prime numbers: 31,37,41,43,47            since there are 5 prime numbers are there between 30 and 50. 3. What are the prime numbers between 70 and 100?

For this I am not writing any formula , simply  with the help of table we will try to understand the concept. the minute and hour hands of a clock form a straight line but in opposite direction 1 Between 12.30 to 12.35 2 Between 1.35 to 1.40 3 Between 2.40 to 2.45 4 Between 3.45 to 3.50 5 Between 4.50 to 4.55 6 Between 5.55 to 6.00  : means at 6.00 clock they are straight but opposite @ Between 6.00 to 6.05 hands are not straight in opposite direction 7  Between 7.05 to 7.10 8 Between  8.10 to 8.15 9 Between  9.15 to 9.20 10 Between  10.20 to 10.25 11 Between  11.25  to 11.30                  #  goto step 1 @ in 6 th column if we see in 5 to 6 hours and 6.00 to 7.00 hours they will straight but opposite only one time. By seeing the above table the minute hand and hour hand s of a clock form straigh