The multiples of 5 are many, in fact, there are an infinite number of them. For example there are the numbers 10, 20 and 35.
The interesting thing is to be able to find a basic and simple rule that allows you to quickly identify if a number is a multiple of 5 or not.
If you look at the multiplication table of 5, taught at school, you can see a certain peculiarity in the numbers on the right.
All results end in 0 or 5, that is, the ones digit is 0 or 5. This is the key to determining whether or not a number is a multiple of 5.
Multiples of 5
Mathematically, a number is a multiple of 5 if it can be written as 5 * k, where "k" is an integer.
Thus, for example, it can be seen that 10 = 5 * 2 or that 35 is equal to 5 * 7.
Since in the previous definition it was said that «k» is an integer, it can also be applied for negative integers, for example for k = -3, we have that -15 = 5 * (- 3) which implies that - 15 is a multiple of 5.
Hence, by choosing different values for "k", different multiples of 5 will be obtained. As the number of integers is infinite, then the number of multiples of 5 will also be infinite.
Euclid's division algorithm
Euclid's Division Algorithm which says:
Given two integers "n" and "m", with m ≠ 0, there are integers "q" and "r" such that n = m * q + r, where 0≤ r <q.
"N" is called a dividend, "m" is called a divisor, "q" is called a quotient, and "r" is called the remainder.
When r = 0 it is said that "m" divides "n" or, equivalently, that "n" is a multiple of "m".
Therefore, wondering what the multiples of 5 are is equivalent to wondering which numbers are divisible by 5.
Because S
Given any integer "n", the possible figures for its unit are any number between 0 and 9.
Looking in detail at the division algorithm for m = 5, it is obtained that «r» can take any of the values 0, 1, 2, 3 and 4.
At the beginning it was concluded that any number when multiplied by 5, will have in the units the figure 0 or the figure 5. This implies that the number of the units of 5 * q is equal to 0 or 5.
Thus, if the sum n = 5 * q + r is carried out, the number of the units will depend on the value of «r» and the following cases exist:
-If r = 0, then the number of the units of «n» is equal to 0 or 5.
-If r = 1, then the number of the units of «n» is equal to 1 or 6.
-If r = 2, then the number of the units of «n» is equal to 2 or 7.
-If r = 3, then the number of the units of «n» is equal to 3 or 8.
-If r = 4, then the number of the units of «n» is equal to 4 or 9.
The above tells us that if a number is divisible by 5 (r = 0), then the number of its units is equal to 0 or 5.
In other words, any number that ends in 0 or 5 will be divisible by 5, or what is the same, it will be a multiple of 5.
For this reason it is only necessary to see the number of the units.
References
- Álvarez, J., Torres, J., lópez, J., Cruz, E. d., & Tetumo, J. (2007). Basic mathematics, supporting elements. Univ. J. Autónoma de Tabasco.
- Barrantes, H., Díaz, P., Murillo, M., & Soto, A. (1998). Introduction to Number Theory. EUNED.
- Barrios, AA (2001). Mathematics 2nd. Editorial Progreso.
- Goodman, A., & Hirsch, L. (1996). Algebra and trigonometry with analytical geometry. Pearson Education.
- Ramírez, C., & Camargo, E. (sf). Connections 3. Editorial Norma.
- Zaragoza, AC (sf). Number theory Editorial Vision Libros.