Divisibility - 3

Category: Arithmetic

"Published in Newark, California, USA"

Divisibility by 3:

How do you know that a number is divisible by 3? Well, a number is divisible by 3 if the sum of the digits are divisible by 3 or multiples of 3. You need to remember or memorize the multiplication table for this one.

Example 1:

The first thing that we need to do is to inspect the given number if it is divisible by 3 or not. Let's add the digits as follows

You can add the digits again as follows

Since 4 is not a multiple of 3, then the given number is not divisible by 3. There's a remainder of 1 in the division. The answer or a quotient must be a whole number itself. You should not have a fraction or a remainder in the final answer.

Example 2:

The first thing that we need to do is to inspect the given number if it is divisible by 3 or not. Let's add the digits as follows

You can add the digits again as follows

 

Since 9 is a multiple of 3, then the given number is divisible by 3. There's no remainder or a fraction in the division.

You should consider in studying the divisibility of a number because you will use these principles later when you will study higher Math subjects that involves the division of a number, simplifying fractions, and even factoring. 

This method can also be used for negative integers as long as the sum of the digits is a multiple of 3. Again, there should be no remainder or a fraction in the division. 
 

Divisibility - 1, 2

Category: Arithmetic

"Published in Newark, California, USA"

Divisibility by 1:

How do you know that a number is divisible by 1? Well, any number divided by 1 is the same number. 

Example 1:

Example 2:

Divisibility by 2:

How do you know that a number is divisible by 2? Well, a number is divisible by 2 if a number ends with 0, 2, 4, 6, and 8. In short, all even numbers are divisible by 2.

Example 1:

Since 585 is not an even number, then it is not divisible by 2. The answer or a quotient must be a whole number itself. You should not have a fraction or a remainder in the final answer. 

Example 2:

Since 3006 is an even number, then it is divisible by 2. The answer or a quotient is a whole number and there's no fraction or remainder.

You should consider in studying the divisibility of a number because you will use these principles later when you will study higher Math subjects that involves the division of a number, simplifying fractions, and even factoring.

By the way, we can do the division of a number without using a calculator or even a scratch paper and a pen. Let's consider this number

Let's start with 5 (left side). 5 divided by 2 is 2. We cannot say 3 because 2 times 3 is 6. 2 times 2 is 4 and that's the highest number that we can consider. 5 minus 4 is 1 and there's a remainder of 1 from the first digit. 

Next, consider the next digit which is 8. Since you have a remainder of 1 from the first digit, then 8 becomes 18. 18 divided by 2 is 9. That's good that 18 is an even number or else we will have another remainder. When you divide any odd numbers by 2, the remainder is always 1.

Finally, consider the last digit which is 5. Since the second digit has no remainder, then we can use 5 in the division. 5 divided by 2 is 2. We cannot say 3 because 2 times 3 is 6. 2 times 2 is 4 and that's the highest number that we can consider. 5 minus 4 is 1. Therefore, the final answer is 292 and has a remainder of 1. 1 is the numerator in the fraction which is the remainder and 2 is the denominator which is the divisor. 

Let's consider another number

Let's start with 3. 3 divided by 2 is 1. We cannot say 2 because 2 times 2 is 4. 2 times 1 is 2 and that's the highest number that we can consider. 3 minus 2 is 1 and there's a remainder of 1 from the first digit.

Next, consider the next digit which is 0. Since you have a remainder of 1 from the first digit, then 0 becomes 10. 10 divided by 2 is 5. That's good that 10 is an even number.

Next, consider the next digit which is 0. Since there's no remainder in the second digit, then we can use 0 in the division. 0 divided by 2 is 0. Zero divided by any number (except zero) is always equal to zero.

Finally, consider the last digit which is 6. Since there's no remainder in the third digit, then we can use 6 in the division. 6 divided by 2 is 3. That's good that 6 is an even number. Therefore, the final answer is 1503. The final answer is a whole number and has no remainder or a fraction.

More Cube Problems, 8

Category: Solid Geometry

"Published in Newark, California, USA"

Show that (a) the total surface of a cube is twice the square of its diagonal, (b) the volume of a cube is 1/9 √3 times the cube of its diagonal.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

Consider the lower base of a cube. Use Pythagorean Theorem in order to solve for the hypotenuse or the diagonal of the lower base as follows

If all the faces of a cube are perpendicular to each other, then all edges are perpendicular to each other also. Since c is located at the lower base of a cube, then c is perpendicular to x. 

Use Pythagorean Theorem in order to solve for the diagonal of a cube as follows

The total area of a cube is

The volume of a cube is

Therefore, 

(a)

(b)