Question

The multiples of which prime number from 1 to 11 will never end with 7 in the ones place. Write down the multiples of the Prime Number.​

Answer 1

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Prime and composite numbers: We can build 36 from 9 and 4 by multiplying; or we can build it from 6 and 6; or from 18 and 2; or even by multiplying 2 × 2 × 3 × 3. Numbers like 10 and 36 and 49 that can be composed as products of smaller counting numbers are called composite numbers.

Some numbers can’t be built from smaller pieces this way. For example, he only way to build 7 by multiplying and by using only counting numbers is 7 × 1. To “build” 7, we must use 7! So we’re not really composing it from smaller building blocks; we need it to start with. Numbers like this are called prime numbers.

Informally, primes are numbers that can’t be made by multiplying other numbers. That captures the idea well, but is not a good enough definition, because it has too many loopholes. The number 7 can be composed as the product of other numbers: for example, it is 2 × 3\frac{1}{2}. To capture the idea that “7 is not divisible by 2,” we must make it clear that we are restricting the numbers to include only the counting numbers: 1, 2, 3….

A formal definition

A prime number is a positive integer that has exactly two distinct whole number factors (or divisors), namely 1 and the number itself.

Clarifying two common confusions

Two common confusions:

The number 1 is not prime.

The number 2 is prime. (It is the only even prime.)

The number 1 is not prime. Why not?

Well, the definition rules it out. It says “two distinct whole-number factors” and the only way to write 1 as a product of whole numbers is 1 × 1, in which the factors are the same as each other, that is, not distinct. Even the informal idea rules it out: it cannot be built by multiplying other (whole) numbers.

But why rule it out?! Students sometimes argue that 1 “behaves” like all the other primes: it cannot be “broken apart.” And part of the informal notion of prime — we cannot compose 1 except by using it, so it must be a building block — seems to make it prime. Why not include it?

Mathematics is not arbitrary. To understand why it is useful to exclude 1, consider the question “How many different ways can 12 be written as a product using only prime numbers?” Here are several ways to write 12 as a product but they don’t restrict themselves to prime numbers.

3 × 4

4 × 3

1 × 12

1 × 1 x 12

2 × 6

1 × 1 × 1 × 2 × 6

Using 4, 6, and 12 clearly violates the restriction to be “using only prime numbers.” But what about these?

3 × 2 × 2

2 × 3 × 2

1 × 2 × 3 × 2

2 × 2 × 3 × 1 × 1 × 1 × 1

Well, if we include 1, there are infinitely many ways to write 12 as a product of primes. In fact, if we call 1 a prime, then there are infinitely many ways to write any number as a product of primes. Including 1 trivializes the question. Excluding it leaves only these cases:

3 × 2 × 2

2 × 3 × 2

2 × 2 × 3

This is a much more useful result than having every number be expressible as a product of primes in an infinite number of ways, so we define prime in such a way that it excludes 1.

The number 2 is prime. Why?

Students sometimes believe that all prime numbers are odd. If one works from “patterns” alone, this is an easy slip to make, as 2 is the only exception, the only even prime. One proof: Because 2 is a divisor of every even number, every even number larger than 2 has at least three distinct positive divisors.

Another common question: “All even numbers are divisible by 2 and so they’re not prime; 2 is even, so how can it be prime?” Every whole number is divisible by itself and by 1; they are all divisible by something. But if a number is divisible only by itself and by 1, then it is prime. So, because all the other even numbers are divisible by themselves, by 1, and by 2, they are all composite (just as all the positive multiples of 3, except 3, itself, are composite).