To find the exponents of prime factors, first perform prime factorization by repeatedly dividing a number by the smallest prime numbers until you're left with only primes; then, count how many times each prime factor appears and write that count as the exponent for that prime base, like 2 × 2 × 2 2 × 2 × 2 becomes 2 3 2 3 .
Prime factorization is the breaking down of a number into the prime numbers that multiply to the original number. For example, the prime factorization of 12 is 2 * 2 * 3. We can add exponents when we have the same prime number occurring more than once. So, the prime factorization of 12 can also be written as 22 * 3.
Draw a circle around a prime number when it appears at the end of a branch. This branch is then complete and no further factors can be found. The tree is complete when all of the branches end with a circled prime number. Then write these prime numbers as a product (multiplication) and use powers to simplify.
For example, 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5. We get the prime factorizations of 60 as 22 × 3 × 5. Just add one (1) to the exponents 2, 1 and 1 individually and multiply their sums. (2 + 1) × (1 + 1) × (1 + 1) = 3 × 2 × 2 = 12.
So, the prime factors of 64 are written as 2 x 2 × 2 x 2 x 2 x 2 or 26, where 2 is a prime number. It is possible to find the exact number of factors of a number 64 with the help of prime factorisation. The prime factor of the 64 is 26. The exponent in the prime factorisation is 6.
Division Method
Step 1: Divide the given number by the smallest prime number. In this case, the smallest prime number should divide the number exactly. Step 2: Again, divide the quotient by the smallest prime number. Step 3: Repeat the process, until the quotient becomes 1.
The number 12,345,678,910,987,654,321 is indeed prime. It consists of 20 digits and is really easy to remember: count to 10 and then count backward again until you get to 1. But it has been unclear whether other primes take the palindromic form of starting at 1, ascending to the number n and then descending again.
Thus, the prime factorization of 1771 is: 1771=7×11×23.
Due to the superstitious significance of the numbers it contains, the palindromic prime 1000000000000066600000000000001 is known as Belphegor's Prime, named after Belphegor, one of the seven princes of Hell.
Speed Trick or Vedic Shortcut
For any number n > 5: If n ends in 0, 2, 4, 5, 6, 8 (even or 5), it's not prime (except 2 & 5). If sum of digits of n is divisible by 3, it's not prime (except 3). Express n as 6k ± 1 (for k an integer): If not, skip divisibility checking.
The exponent of any prime in a factorial is given by, e1 = [n/p1] + [n/p12] + [n/p13] … The power of 5 in 100! is 24.
7 Rules for Exponents with Examples
So, the prime factors of 32 are written as 2 × 2 x 2 x 2 x 2 or 25, where 2 is a prime numbers. It is possible to find the exact number of factors of a number with the help of prime factorisation. The prime factor of the 32 is 25. The exponent in the prime factorisation is 5.
The sum of exponents of prime factors in the primefactorisation of 1764 is (a) 3 (b) 4 (c) 5 (d) 6 Ans : Prime factors of 1764 , 1764=4×49×9=22×72×32 The sum of exponents of prime factor is 2+2+2=6.
So, the prime factors of 54 are written as 2 × 3 × 3 × 3 or 2x 33, where 2 and 3 are the prime numbers.
Answer: The prime factorisation of 34 is equal to 2 x 17. Therefore, 2 and 17 are the prime factors of 34.
The number 2099 has only two factors, 1 and 2099, so it meets the definition of a prime number.
No, 111 is not a prime number. The number 111 is divisible by 1, 3, 37, 111. For a number to be classified as a prime number, it should have exactly two factors. Since 111 has more than two factors, i.e. 1, 3, 37, 111, it is not a prime number.
Factoring Methods
There are overall 8 factors of 999 among which 999 is the biggest factor and its positive factors are 1, 3, 9, 27, 37, 111, 333, 999. The Prime Factors and Pair Factors of 999 are 3 × 37 and (1, 999), (3, 333), (9, 111), (27, 37) respectively.