Question

Observe that \[2 \times 3 + 1\] = \[7\] is a prime number . Here \[1\] has been added to a multiple of \[2\] to get a prime number . Can you find some more numbers of this type ?

Answer 1

Hint: First we have to remember which numbers are the prime numbers . Prime numbers are the numbers which are not divisible by any other numbers except \[1\] and the number itself . So we can say prime numbers have total \[2\] factors .We will check all the numbers to know which numbers are prime numbers and how we can express them \[1\] in addition to any multiple of \[2\] .

Complete step-by-step answer:
Now we will see which numbers are prime numbers and we can write each prime number except \[2\] as \[1\] in addition to any multiple of \[2\] .
\[1\] is not a prime number . It has only one factor .
\[2\]is a prime number . It has two factors \[1\] and itself.
\[3\] is a prime number . It has two factors \[1\] and itself. \[3 = 2 \times 1 + 1\]
\[4\]is not a prime factor . It has three factors \[1\],\[2\]and itself.
\[5\] is a prime number . It has two factors \[1\] and itself. \[5 = 2 \times 2 + 1\]
\[6\] is not a prime factor . It has three factors \[1\],\[2\],\[3\]and itself.
\[7\] is a prime number . It has two factors \[1\] and itself. \[7 = 2 \times 3 + 1\]
\[8\] is not a prime factor . It has three factors \[1\],\[2\],\[4\]and itself.
\[9\] is not a prime factor . It has three factors \[1\] ,\[3\]and itself.
\[10\] is not a prime factor . It has three factors \[1\],\[2\],\[5\]and itself.
\[11\] is a prime number . It has two factors \[1\] and itself. \[11 = 2 \times 5 + 1\]
\[12\]is not a prime factor . It has three factors \[1\],\[2\],\[3\],\[4\], \[6\]and itself.
\[13\] is a prime number . It has two factors \[1\] and itself. \[13 = 2 \times 6 + 1\]
\[14\] is not a prime factor . It has three factors \[1\],\[2\], \[7\] and itself.
\[15\] is not a prime factor . It has three factors \[1\],\[3\], \[5\] and itself.
\[16\] is not a prime factor . It has three factors \[1\], \[2\] ,\[4\], \[8\] and itself.
\[17\] is a prime number . It has two factors \[1\] and itself. \[17 = 2 \times 8 + 1\]
\[18\] is not a prime factor . It has three factors \[1\], \[2\] , \[3\] , \[6\], \[9\] and itself.
\[19\] is a prime number . It has two factors \[1\] and itself . \[19 = 2 \times 9 + 1\]
\[20\] is not a prime factor . It has three factors \[1\], \[2\] ,\[4\], \[5\] , \[10\] and itself.
Other Prime numbers are \[23\], \[29\], \[31\], \[37\] etc .
\[23\] can be written as \[23 = 2 \times 11 + 1\]
\[29\] can be written as \[29 = 2 \times 14 + 1\]
\[31\] can be written as \[31 = 2 \times 15 + 1\]
\[37\]can be written as \[37 = 2 \times 18 + 1\]
So we can see that we can find some more numbers of the type \[2 \times 3 + 1\] = \[7\].
They are \[2 \times 1 + 1 = 3\], \[2 \times 2 + 1 = 5\], \[2 \times 5 + 1 = 11\], \[2 \times 6 + 1 = 13\], \[2 \times 8 + 1 = 17\], \[2 \times 9 + 1 = 19\], \[2 \times 11 + 1 = 23\], \[2 \times 14 + 1 = 29\],\[2 \times 15 + 1 = 31\],\[2 \times 18 + 1 = 37\] and so on .

Note: Not all odd numbers are prime numbers . We get every odd number by adding \[1\] to multiple of \[2\]. All prime numbers are odd . So every prime number is \[1\] in addition to multiple of \[2\] but not \[1\] in addition to every multiple of \[2\] is prime number . Like \[2 \times 4 + 1 = 9\], \[2 \times 7 + 1 = 15\] are not prime numbers .