Three different prime numbers have to be involved in this question. We should choose them to make life as easy as possible for ourselves: we don't want to be working with big numbers. The three smallest prime numbers are 2, 3 and 5. 5 is the largest, so we will give it the lowest exponent. 52 = 25 33 = 27 24 = 16 This means that we are looking for a multiple of 25, n, such that n+1 divides exactly by 27 and n+2 by 16. In other words, n/27 leaves remainder 26, and n/16 leaves remainder 14. When we divide multiples of 25 by 27, the remainders follow a cycle like this: [0], 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, 26, 24, 22, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0... Notice that the cycle is itself of length 27. That is because 27 and 25 do not share any prime factors: you have to multiply 25 by 27 before you get a number that is a multiple of both. When we divide multiples of 25 by 16, the remainders follow a cycle like this: [0], 9, 2, 11, 4, 13, 6, 15, 8, 1, 10, 3, 12, 5, 14, 7, 0... It is our good fortune that the desired remainders occur in 14th position in both series simultaneously, which means we don't need to look any farther. Our value of n is 14 x 25. n = 350 = 5 x 5 x 2 x 7 n + 1 = 351 = 3 x 3 x 3 x 13 n + 2 = 352 = 2 x 2 x 2 x 2 x 2 x 11 Our remainder series aligned very nicely to give us a small answer, at n = 14 x 25. The next valid value of n is when they next align at: n = [(27 x 16) + 14] x 25 = 11150. However, that is using criteria that we specified as we went along. There are likely to be many other smallish solutions to the original question.
9 is not a prime factor, but it does have one prime factor, which is three. Prime factors are factors of a number that are prime.
Three.
The smallest prime factor of 195 is three.
105
7,000 has three prime factors: 2, 5 and 7.
The number 32 has only one prime factor: 2The true prime factor for any number is 1.
Yes. Any time three is a factor, it is a prime factor.
A prime number is an integer, a, that can only be expressed as a * 1 = a; a > 0. It has no other factors save for 1 and itself. Thus, the only common factor of 2 or more prime numbers is 1.
Not a lot of room between 29 and 30. Prime factorization is the expression of a positive composite integer as the product of its prime factors. 30 has three prime factors. 2 x 3 x 5 is the prime factorization of 30.
9 is not a prime factor, but it does have one prime factor, which is three. Prime factors are factors of a number that are prime.
30 has three prime numbers and is a factor of 60.
Three.
15 is not a prime number, so it can't be a prime factor.
Both are prime numbers
The smallest prime factor of 195 is three.
No. 127 is a prime number
All composite numbers can be expressed as unique products of prime numbers. This is accomplished by dividing the original number and its factors by prime numbers until all the factors are prime. A factor tree can help you visualize this. Example: 210 210 Divide by two. 105,2 Divide by three. 35,3,2 Divide by five. 7,5,3,2 Stop. All the factors are prime. 2 x 3 x 5 x 7 = 210 That's the prime factorization of 210.