Seventy Eight (78)
1. It is the smallest number which can be written as the sum of four distinct squares in 3 ways.
78 = 1^2 + 2^2 +3^2 + 8^2
= 1^2 + 4^2 + 5^2 + 6^2
= 2^2 + 3^2 + 4^2 + 7^2
2. It is a triangular number.
3. It is the sum of first 12 natural numbers.
1 + 2 + 3 + --- + 12 = 78
4. There are 78 Tarot cards which is used for fortune telling. 56 among them the cards of lesser arcana and 22 are of greater arcana.
5. It can be written as the sum of two consecutive primes.
78 = 37 + 41
6. It is a 27-gonal number.
7. It is divisible by three distinct prime number 2, 3 and 13 which makes it a member of Sphenic number. A Sphenic number is a positive integer having three distinct prime factors.
The first five Sphenic numbers are – 30, 42, 66, 70 and 78.
8. The atomic number of Platinum is 78.
9. The number of chromosomes in canine of DNA is 78.
10. It is an abundant number.
Rajesh Kumar Thakur
rkthakur1974@gmail.com
Sunday, February 7, 2016
Matheamatically 77
Seventy Seven (77)
1. It is the product of two odd primes.
77 = 7 x 11
2. It is the sum of first 8 prime numbers.
77 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19
3. Any number that is based on the pattern abcabc will be divisible by 77.
123123 ÷ 77 = 1599
4. It is the largest number that cannot be written as the sum of distinct numbers whose reciprocals add up to 1.
78 = 2 + 6 + 8 + 10 + 12 + 40
1 = ½ + 1/6 + 1/8 + 1/10 + 1/12 + 1/40
5. FORTRAN 77 was a computer programming developed by IBM in 1950.
6. It can be expressed as the sum of squares of three consecutive numbers.
77 = 4^2 + 5^2 + 5^2
7. The atomic number of Iridium is 77.
8. It can be partitioned 39 times with each tern not larger than 2 and 533 times with each term no larger than 2.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the product of two odd primes.
77 = 7 x 11
2. It is the sum of first 8 prime numbers.
77 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19
3. Any number that is based on the pattern abcabc will be divisible by 77.
123123 ÷ 77 = 1599
4. It is the largest number that cannot be written as the sum of distinct numbers whose reciprocals add up to 1.
78 = 2 + 6 + 8 + 10 + 12 + 40
1 = ½ + 1/6 + 1/8 + 1/10 + 1/12 + 1/40
5. FORTRAN 77 was a computer programming developed by IBM in 1950.
6. It can be expressed as the sum of squares of three consecutive numbers.
77 = 4^2 + 5^2 + 5^2
7. The atomic number of Iridium is 77.
8. It can be partitioned 39 times with each tern not larger than 2 and 533 times with each term no larger than 2.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 76
Seventy Six (76)
1. It is a Lucas number. The sequence in Lucas also moves like Fibonacci sequence. The numbers in Lucas series are- 1, 3, 4, 7, 11, 18, 29, 47, 76..
2. Sum of 4 consecutive Fibonacci sequence is 76.
8 + 13 + 21 + 34 = 76
3. It can be expressed as the sum of squares of primes.
76 = 32 + 32 + 32 + 72
4. Any power of 76 end with 76. Such numbers whose any power ends with the number itself is called automorphic. There are only two automorphic numbers 25 and 76 that are below 100.
76^2 = 5776 76^3 = 438976
76^4 = 33362176 76^5 = 2535525376
5. The atomic number of Osmium is 76.
6. It is a 14-gonal number.
7. It is a centered 25-gonal number.
8. It is a centered 25-gonal number.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a Lucas number. The sequence in Lucas also moves like Fibonacci sequence. The numbers in Lucas series are- 1, 3, 4, 7, 11, 18, 29, 47, 76..
2. Sum of 4 consecutive Fibonacci sequence is 76.
8 + 13 + 21 + 34 = 76
3. It can be expressed as the sum of squares of primes.
76 = 32 + 32 + 32 + 72
4. Any power of 76 end with 76. Such numbers whose any power ends with the number itself is called automorphic. There are only two automorphic numbers 25 and 76 that are below 100.
76^2 = 5776 76^3 = 438976
76^4 = 33362176 76^5 = 2535525376
5. The atomic number of Osmium is 76.
6. It is a 14-gonal number.
7. It is a centered 25-gonal number.
8. It is a centered 25-gonal number.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Sunday, January 31, 2016
Mathematically 75
Seventy Five (75)
1. It is also called three quarter. 3/ 4 = 75%
2. It is one less than the sum of squares of its digit.
75 = 7^2 + 5^2 – 1
3. It can be expressed as the sum of 10 consecutive primes beginning from 3.
75 = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
4. It is a nonagonal number.
5. It is a pentagonal pyramid number. A pentagonal pyrammdal number is a figurarive number that represents the number of objects in a pyramid with pentagonal base.
75 = 1 + 6 + 18 + 40
6. The atomic number of Rbenium is 75.
7. It is a Keith number. A Keith numbers run like a Fibonacci sequence. 75 has its digit 7 and 5. If we start to form a sequence with the digit 75 and go on finding the next number by adding the previous two as done in Fibonacci sequence we get 75.
7, 5 , 12, 17, 29, 46, 75.. It is named after Mike Keith who discovered this number in 1987
8. It is a Self-number. A self-number is an integer which in a given base cannot be generated by any other integer added to the sum of that other integer’s digit.
Example:- 21 is not a self-number as it can be generated by 15 and sum of tis digit 1 and 5, whereas 20 is a self-number as it cannot be generated by any such numbers.
9. A diamond anniversary is celebrated after every 75 years.
10. The Proper divisor of 75 are - 1, 3, 5, 15, 25 whose sum is a perfect square
1 + 3 + 5 + 15 + 25 = 49 = 7^2
Moreover , its product
1 x 3 x 5 x 15 x 25 = 5625 = 75^2
is also a square.
11. 2^75 + 75 is a prime
Dr Rajesh Kumar Thakur
1. It is also called three quarter. 3/ 4 = 75%
2. It is one less than the sum of squares of its digit.
75 = 7^2 + 5^2 – 1
3. It can be expressed as the sum of 10 consecutive primes beginning from 3.
75 = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
4. It is a nonagonal number.
5. It is a pentagonal pyramid number. A pentagonal pyrammdal number is a figurarive number that represents the number of objects in a pyramid with pentagonal base.
75 = 1 + 6 + 18 + 40
6. The atomic number of Rbenium is 75.
7. It is a Keith number. A Keith numbers run like a Fibonacci sequence. 75 has its digit 7 and 5. If we start to form a sequence with the digit 75 and go on finding the next number by adding the previous two as done in Fibonacci sequence we get 75.
7, 5 , 12, 17, 29, 46, 75.. It is named after Mike Keith who discovered this number in 1987
8. It is a Self-number. A self-number is an integer which in a given base cannot be generated by any other integer added to the sum of that other integer’s digit.
Example:- 21 is not a self-number as it can be generated by 15 and sum of tis digit 1 and 5, whereas 20 is a self-number as it cannot be generated by any such numbers.
9. A diamond anniversary is celebrated after every 75 years.
10. The Proper divisor of 75 are - 1, 3, 5, 15, 25 whose sum is a perfect square
1 + 3 + 5 + 15 + 25 = 49 = 7^2
Moreover , its product
1 x 3 x 5 x 15 x 25 = 5625 = 75^2
is also a square.
11. 2^75 + 75 is a prime
Dr Rajesh Kumar Thakur
Matheamatically 74
Seventy Four (74)
1. The maximal number of regions into which a plane can be divided by 9 circles are 74.
2. It is the product of two distinct prime.
74 = 2 x 37
3. It is the sum of four consecutive numbers.
74 = 17 + 18 + 19 + 20
4. The atomic number of Tungsten is 74.
Regards
Dr Rajesh Kumar Thakur
1. The maximal number of regions into which a plane can be divided by 9 circles are 74.
2. It is the product of two distinct prime.
74 = 2 x 37
3. It is the sum of four consecutive numbers.
74 = 17 + 18 + 19 + 20
4. The atomic number of Tungsten is 74.
Regards
Dr Rajesh Kumar Thakur
Mathematically 73
Seventy Three (73)
1. It is the 21st prime number. It along with the next prime 73 makes a twin prime.
2. It is the smallest number besides 1 which is 1 less than the double of its reverse.
73 = 2 x 37 – 1
3. It is the smallest number with the property that its neighbours can be written as the sum of two squares.
72 = 6^2 + 6^2
73 = 3^2 + 8^2
74 = 5^2 + 8^2
4. It is the smallest prime congruent to 1 modulo 24.
73 ≡ 1 (mod 24)
5. On base 8 it is the only number with repeated unit.
73 = ( 111 ) 8
6. The square root of 73 is 8. 544.. which when summed up gives 21 that is the product of its digit.
(73)^1/2 = 8. 544 and 8 + 5 + 4 + 4 = 7 x 3
7. It is a centered 12 gonal and 24 gonal number.
8. It is the least number of sixth powers needed to represent every possible integer.
9. The atomic number of Tantalum is 73.
10. There is a very interesting puzzle whose final result comes out to be 73.
The steps of the puzzles are--
-a) Take any four digit number
b) Write this number again making a eight digit number
c) Divide this 8 digit number by 137
d) divide the quotient by the 4 digit number taken initially.
The final result will be 73.
Example: - Let the four digit number taken is 1238.
Make this number an eight digit number by writing this number twice, i.e. 12381238.
Divide this big number by 137 and you get the quotient 90374 which when divided by 1238 gives you 73.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the 21st prime number. It along with the next prime 73 makes a twin prime.
2. It is the smallest number besides 1 which is 1 less than the double of its reverse.
73 = 2 x 37 – 1
3. It is the smallest number with the property that its neighbours can be written as the sum of two squares.
72 = 6^2 + 6^2
73 = 3^2 + 8^2
74 = 5^2 + 8^2
4. It is the smallest prime congruent to 1 modulo 24.
73 ≡ 1 (mod 24)
5. On base 8 it is the only number with repeated unit.
73 = ( 111 ) 8
6. The square root of 73 is 8. 544.. which when summed up gives 21 that is the product of its digit.
(73)^1/2 = 8. 544 and 8 + 5 + 4 + 4 = 7 x 3
7. It is a centered 12 gonal and 24 gonal number.
8. It is the least number of sixth powers needed to represent every possible integer.
9. The atomic number of Tantalum is 73.
10. There is a very interesting puzzle whose final result comes out to be 73.
The steps of the puzzles are--
-a) Take any four digit number
b) Write this number again making a eight digit number
c) Divide this 8 digit number by 137
d) divide the quotient by the 4 digit number taken initially.
The final result will be 73.
Example: - Let the four digit number taken is 1238.
Make this number an eight digit number by writing this number twice, i.e. 12381238.
Divide this big number by 137 and you get the quotient 90374 which when divided by 1238 gives you 73.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 72
Seventy Two (72)
1. It is equal to half of the gross.
½ x 144 = 72
2. The sum of first eight even number is 72.
2 + 4 + 6 + 8 +10 + 12 + 14 + 16 = 72
3. It is the only number which is eight times the sum of its own digits.
72 = 8 x (7 + 2)
4. It is the smallest number that can be written as the sum of two prime number in two distinct ways and the most interesting part of this partition is that the primes used ends with 1.
72 = 11 + 61
= 31 + 41
5. The normal human pulse rate is 72 beats/ minutes at rest.
6. An isosceles triangle having a pair of 72 degree angle is called Golden triangle.
7. It is one- fifth of a central angle (360 degree) of circle.
8. The internal angle of a regular pentagon is 72 degree.
9. It is divisible by every number from 1 to 10 apart from 5, 7 and 10.
10. It is the smallest 5th power equal to the sum of 5 other 5th powers.
72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
11. In financial circle rule of 72 is a simple method to quickly estimate compound interest. If you want to know how long will it take to double your money, divide 72 by the interest rate and the answer will be the number of years.
12. It is the sum of four consecutive primes as well as the sum of six consecutive prime number.
72 = 13 + 17 + 19 + 23 (sum of 4 consecutive numbers)
= 5 + 7 + 11 + 13 + 17 + 19 (sum of 6 consecutive numbers)
13. It is a Harshad number on base 10.
14. The atomic number of Hafnium is 72.
15. It is an Abundant number because the sum of factors of 72 excluding itself is more than 72. The factors are- 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
16. It is a 25- gonal number.
17. It is the maximum number of sphere that can touch another sphere in a lattice packing in 6 dimensions.
18. The great Rhombicuboctahedron in an Archimedean solid is 72 edges.
19. The sum of the eight row of Lozanic triangle is 72.
72 = 1 + 4 + 12 + 19 + 19 + 12 + 4 + 1
The Lozanic triangle is a triangular array of binomial coefficients similar to Pascal’s triangle. It is named after Sima Lozanic, the Serbian chemist.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is equal to half of the gross.
½ x 144 = 72
2. The sum of first eight even number is 72.
2 + 4 + 6 + 8 +10 + 12 + 14 + 16 = 72
3. It is the only number which is eight times the sum of its own digits.
72 = 8 x (7 + 2)
4. It is the smallest number that can be written as the sum of two prime number in two distinct ways and the most interesting part of this partition is that the primes used ends with 1.
72 = 11 + 61
= 31 + 41
5. The normal human pulse rate is 72 beats/ minutes at rest.
6. An isosceles triangle having a pair of 72 degree angle is called Golden triangle.
7. It is one- fifth of a central angle (360 degree) of circle.
8. The internal angle of a regular pentagon is 72 degree.
9. It is divisible by every number from 1 to 10 apart from 5, 7 and 10.
10. It is the smallest 5th power equal to the sum of 5 other 5th powers.
72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
11. In financial circle rule of 72 is a simple method to quickly estimate compound interest. If you want to know how long will it take to double your money, divide 72 by the interest rate and the answer will be the number of years.
12. It is the sum of four consecutive primes as well as the sum of six consecutive prime number.
72 = 13 + 17 + 19 + 23 (sum of 4 consecutive numbers)
= 5 + 7 + 11 + 13 + 17 + 19 (sum of 6 consecutive numbers)
13. It is a Harshad number on base 10.
14. The atomic number of Hafnium is 72.
15. It is an Abundant number because the sum of factors of 72 excluding itself is more than 72. The factors are- 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
16. It is a 25- gonal number.
17. It is the maximum number of sphere that can touch another sphere in a lattice packing in 6 dimensions.
18. The great Rhombicuboctahedron in an Archimedean solid is 72 edges.
19. The sum of the eight row of Lozanic triangle is 72.
72 = 1 + 4 + 12 + 19 + 19 + 12 + 4 + 1
The Lozanic triangle is a triangular array of binomial coefficients similar to Pascal’s triangle. It is named after Sima Lozanic, the Serbian chemist.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
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