One Hundred (100)
1. It is the smallest three digit number.
2. It is the sum of first 10 odd numbers.
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100
3. Boiling point of water is 100^0C
4. It is the sum of 2 consecutive triangular numbers.
45 + 55 = 100
5. It is the sum of 9 consecutive primes.
2 + 3 + 5 + 7 + 11 +13 +17 + 19 + 23 = 100
6. It is a square number.
10^2 = 100
7. It can be expressed as the sum of cube of first four natural numbers.
1^3 + 2^3 + 3^3 + 4^3 = 100
8. It is an abundant number.
9. It is a centered 33 gonal number.
10. It can be expressed as the sum of four pairs of prime numbers.
47 + 53 = 17 + 83 = 3 + 97 = 41 + 59 = 100
11. It is a Harshad number in base 10.
12. It is a Leyland number which can be expressed in the form of a^b + b^a .
100 = 2^6 + 6^2
13. It is the smallest number whose common logarithm is a prime number.
log 10^100 = 2
14. The atomic number of Fermium is 100.
15. A century has 100 years.
16. There is a very interesting puzzle which says – using digits from 1 to 9, and mathematical operators + and - , make 100. There are several solutions to this puzzle few of them are provided here.
98 – 76 + 54 + 3 + 21 =100
9 – 8 + 76 + 54 – 32 + 1 = 100
98 + 7 + 6 – 5 – 4 – 3 + 2 – 1 = 100
9 – 8 + 76 – 5 + 4 + 3 + 21 = 100
98 – 7 – 6 – 5 – 4 + 3 + 21 = 100
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Dr Rajesh Kumar Thakur
Saturday, February 27, 2016
Mathematically 98
Ninety Eight (98)
1. It can be expressed as the fourth power of sum of first three natural numbers.
98 = 1^4 + 2^4 + 3^4
2. The atomic number of Californium is 98.
3. It can be expressed as the difference between two odd consecutive cubes.
98 = 53 – 33
4. The reciprocal of 98 = 1/98 on division shows an interesting pattern of GP, though the patterns discontinues but it starts with the power of 2.
1/98 = 0. 01 02 04 08 16 32 65 30 ….
Here the number shown in bold is in GP.
5. It is the lowest number such that the first 5 multiple of it contains the digit 9.
98 x 1 = 98
98 x 2 = 196
98 x 3 = 294
98 x 4 = 392
98 x 5 = 490
6. It is the smallest number which cannot be expressed as the sum of two primes if the first prime is either of 3, 5 or 7. The next such numbers are 122, 124, 126 and 128.
Dr Rajesh Kumar Thakur
1. It can be expressed as the fourth power of sum of first three natural numbers.
98 = 1^4 + 2^4 + 3^4
2. The atomic number of Californium is 98.
3. It can be expressed as the difference between two odd consecutive cubes.
98 = 53 – 33
4. The reciprocal of 98 = 1/98 on division shows an interesting pattern of GP, though the patterns discontinues but it starts with the power of 2.
1/98 = 0. 01 02 04 08 16 32 65 30 ….
Here the number shown in bold is in GP.
5. It is the lowest number such that the first 5 multiple of it contains the digit 9.
98 x 1 = 98
98 x 2 = 196
98 x 3 = 294
98 x 4 = 392
98 x 5 = 490
6. It is the smallest number which cannot be expressed as the sum of two primes if the first prime is either of 3, 5 or 7. The next such numbers are 122, 124, 126 and 128.
Dr Rajesh Kumar Thakur
Mathematically 97
Ninety Seven (97)
1. It is a 25th prime number and the largest two digit prime number.
2. It is the highest two digit prime number whose cube has no zero.
97^3 = 912673
3. In Georgian calendar there are 97 leap days in every 400 years.
4. It is a 4 dimensional centered cube number.
5. It is a permutable primes. A prime is called permutable if on a given base, the prime always remain a prime despite its digit’s position changed through permutation. It was first studied by H E Richert. 97 and its reverse 79 are primes .The first 10 permutable primes are- 2, 3, 5, 7, 11, 13, 17, 31, 37 and 71.
6. It can be written in the form of – n^4 + (n +1)^4
97 = 2^4 + 3^4
7. The period of reciprocal of 97 = 1/97 is maximum and its length is 96. More interestingly, the first four pairs of its expansion are in Geometric Progression (GP). Alexender Aitken, a professor at Edinburgh University knew it by heart.
1/97 = 0.01 03 09 27 83 50 51 57 52 57 73 19 58 76 28 86 59 79……
8. It is a Proth prime. A Proth prime is written in the form of – k x 2n + 1, where k is an odd positive integer and 2n > k. It is named after the mathematician Francois Proth. The first 10 Proth primes are- 3, 5, 9, 13, 17, 25, 33, 41, 49 and 57.
97 = 3 x 25 + 1
9. The atomic number of Berkelium is 97
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a 25th prime number and the largest two digit prime number.
2. It is the highest two digit prime number whose cube has no zero.
97^3 = 912673
3. In Georgian calendar there are 97 leap days in every 400 years.
4. It is a 4 dimensional centered cube number.
5. It is a permutable primes. A prime is called permutable if on a given base, the prime always remain a prime despite its digit’s position changed through permutation. It was first studied by H E Richert. 97 and its reverse 79 are primes .The first 10 permutable primes are- 2, 3, 5, 7, 11, 13, 17, 31, 37 and 71.
6. It can be written in the form of – n^4 + (n +1)^4
97 = 2^4 + 3^4
7. The period of reciprocal of 97 = 1/97 is maximum and its length is 96. More interestingly, the first four pairs of its expansion are in Geometric Progression (GP). Alexender Aitken, a professor at Edinburgh University knew it by heart.
1/97 = 0.01 03 09 27 83 50 51 57 52 57 73 19 58 76 28 86 59 79……
8. It is a Proth prime. A Proth prime is written in the form of – k x 2n + 1, where k is an odd positive integer and 2n > k. It is named after the mathematician Francois Proth. The first 10 Proth primes are- 3, 5, 9, 13, 17, 25, 33, 41, 49 and 57.
97 = 3 x 25 + 1
9. The atomic number of Berkelium is 97
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 96
Ninety Six (96)
1. It is a sixth star number.
2. It is the smallest number that can be written as the difference of 2 squares in 4 ways.
96 = 252 – 232 = 142 - 102 = 112 – 52 = 102 – 22
3. It can be expressed as the difference of factorials of two consecutive numbers.
96 = 5! – 4! = 120 – 24
4. It is the 7th number that stays same when written upside down.
5. It is the second smallest number with 6 prime factors.
96 = 3 x 2 x 2 x 2 x 2 x 2
6. It is a refactorable number. A number is called refactorable or tau number if it is divisible by the count of its divisors. 96 has 12 divisors – 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96. Such numbers were first studied by Curtis Cooper and Robert E Kennedy and later rediscovered by Simon Colton. The first 10 Tau numbers are- 1, 2, 8, 9, 12, 18, 24, 36, 40 and 56.
7. It is an octagonal number.
8. It is an abundant number.
9. It is a 33 gonal number.
10. The atomic number of Curium is 96.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a sixth star number.
2. It is the smallest number that can be written as the difference of 2 squares in 4 ways.
96 = 252 – 232 = 142 - 102 = 112 – 52 = 102 – 22
3. It can be expressed as the difference of factorials of two consecutive numbers.
96 = 5! – 4! = 120 – 24
4. It is the 7th number that stays same when written upside down.
5. It is the second smallest number with 6 prime factors.
96 = 3 x 2 x 2 x 2 x 2 x 2
6. It is a refactorable number. A number is called refactorable or tau number if it is divisible by the count of its divisors. 96 has 12 divisors – 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96. Such numbers were first studied by Curtis Cooper and Robert E Kennedy and later rediscovered by Simon Colton. The first 10 Tau numbers are- 1, 2, 8, 9, 12, 18, 24, 36, 40 and 56.
7. It is an octagonal number.
8. It is an abundant number.
9. It is a 33 gonal number.
10. The atomic number of Curium is 96.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 95
Ninety Five (95)
1. It is the sum of 7 consecutive prime starting from 5.
95 = 5 + 7 + 11 + 13 + 17 + 19 + 23
2. It can be expressed as the sum of squares of 4 numbers distinctly.
95 = 1^2 + 2^2 + 3^2 + 9^2
3. It is a hexagonal pyramidal number.
4. It is a Thabit number. A number that can be expresses as 3 x 2n – 1 for n > 0 is known as Thabit number. It is named after Iraqi mathematician of 9th century Thabit ibn Qurra. The first ten Thabit numbers are- 2, 5, 11, 23, 47, 95, 191, 383, 767 and 1535.
5. The number of planar partition of 10 is 95.
6. The atomic number of Americium is 95.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the sum of 7 consecutive prime starting from 5.
95 = 5 + 7 + 11 + 13 + 17 + 19 + 23
2. It can be expressed as the sum of squares of 4 numbers distinctly.
95 = 1^2 + 2^2 + 3^2 + 9^2
3. It is a hexagonal pyramidal number.
4. It is a Thabit number. A number that can be expresses as 3 x 2n – 1 for n > 0 is known as Thabit number. It is named after Iraqi mathematician of 9th century Thabit ibn Qurra. The first ten Thabit numbers are- 2, 5, 11, 23, 47, 95, 191, 383, 767 and 1535.
5. The number of planar partition of 10 is 95.
6. The atomic number of Americium is 95.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 94
Ninety Four (94)
1. It is a Smith number. A number is called Smith number if the sum of digits of its prime factor is equal to the sum of the digits of the number.
94 = 2 x 47 and 9 + 4 = 2 + 4 + 7
2. It is a 17- gonal number.
3. Apart from 2 and 4, it is the smallest even number which cannot be written as the sum of two distinct semi primes.
4. It is a centered 31 gonal number.
5. It can be partitioned 48 times with each term not larger than 2 and 784 times with each term not larger than 3.
6. The atomic number of Plutonium is 94. It is the notorious chemical element which doesn’t exist naturally and was first created artificially in 1940.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a Smith number. A number is called Smith number if the sum of digits of its prime factor is equal to the sum of the digits of the number.
94 = 2 x 47 and 9 + 4 = 2 + 4 + 7
2. It is a 17- gonal number.
3. Apart from 2 and 4, it is the smallest even number which cannot be written as the sum of two distinct semi primes.
4. It is a centered 31 gonal number.
5. It can be partitioned 48 times with each term not larger than 2 and 784 times with each term not larger than 3.
6. The atomic number of Plutonium is 94. It is the notorious chemical element which doesn’t exist naturally and was first created artificially in 1940.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 93
Ninety Three (93)
1. With just a straight cut, a potato can be divided into 93 pieces.
2. It is an odd number with the property that the digit at ten’s place is thrice the digit at one’s place.
3. It is a 32 gonal number.
4. It can be partitioned 47 times with each tern not larger than 2 and 768 times where each term is not larger than 3.
5. It is the first member of the triplet of consecutive semi-primes 93, 94 and 95. A number is called semi primes if it can be expressed as the product of two prime numbers.
93 = 3 x 31 94 = 2 x 47 95 = 5 x 19
6. It is a palindrome number on base 2.
(93)10 = (1011101)2
7. On base 5, it is a repeated digit.
(93)10 = (333)5
8. It can be expressed as the sum of squares of three numbers. Interestingly all the three numbers used are in Arithmetic Progression with common difference 3.
93 = 2^2 + 5^2 + 8^2
9. It is the sum of 6 consecutive numbers.
93 = 13 + 14 + 15 + 16 + 17 + 18
10. It is a Blum integer. A natural number n is a Blum integer if n = p x q is a semi prime and p and q are written in the form of 4x + 3. It is named after computer scientist Manuel Blum.
93 = 3 x 31
where 3 and 31 are prime numbers
31 = 4 x 7 + 3
11. The atomic number of Neptunium is 93.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. With just a straight cut, a potato can be divided into 93 pieces.
2. It is an odd number with the property that the digit at ten’s place is thrice the digit at one’s place.
3. It is a 32 gonal number.
4. It can be partitioned 47 times with each tern not larger than 2 and 768 times where each term is not larger than 3.
5. It is the first member of the triplet of consecutive semi-primes 93, 94 and 95. A number is called semi primes if it can be expressed as the product of two prime numbers.
93 = 3 x 31 94 = 2 x 47 95 = 5 x 19
6. It is a palindrome number on base 2.
(93)10 = (1011101)2
7. On base 5, it is a repeated digit.
(93)10 = (333)5
8. It can be expressed as the sum of squares of three numbers. Interestingly all the three numbers used are in Arithmetic Progression with common difference 3.
93 = 2^2 + 5^2 + 8^2
9. It is the sum of 6 consecutive numbers.
93 = 13 + 14 + 15 + 16 + 17 + 18
10. It is a Blum integer. A natural number n is a Blum integer if n = p x q is a semi prime and p and q are written in the form of 4x + 3. It is named after computer scientist Manuel Blum.
93 = 3 x 31
where 3 and 31 are prime numbers
31 = 4 x 7 + 3
11. The atomic number of Neptunium is 93.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
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