Often when solving problems, you need to find out if a particular number is divisible by a given digit without a remainder. But each time to share it for a very long time. In addition, there is a high probability of making an error in the calculations and avoiding the correct answer. In order to avoid this problem, signs of divisibility into basic prime or single-digit numbers were found: 2, 3, 9, 11. But what if we need to divide by another large digit? For example, how to calculate the sign of divisibility by 15? We will try to find the answer to this question in this article.
How to formulate a sign of divisibility by 15?
If for prime numbers the signs of divisibility are well known, then what to do with the rest?
If the number is not prime, then it can be factorized. For example, 33 is the product of 3 and 11, and 45 is the product of 9 and 5. There is a property according to which a number is divided by a given one without a remainder if it can be divided by either factor. This means that any large number can be represented in the form of simple ones, and already proceeding from them, to formulate a sign of divisibility.
So, we need to find out if this number can be divided by 15. To do this, consider it in more detail. The number 15 can be represented as the product of 3 and 5. This means that the number is divisible by 15, it must be a multiple of 3 and 5 at the same time. This is a sign of divisibility by 15. In the future, we will consider it in more detail and formulate it more precisely.
How to find out that a number is divisible by 3?
Recall the sign of divisibility by 3.
A number is divided by 3 if the sum of its numbers (the number of units, tens, hundreds, and so on) is divided by 3.
So, for example, you need to find out which of these numbers can be divided by 3 without a remainder: 76348, 24606, 1128904, 540813.
Of course, you can simply divide the given numbers into a column, but it will take a lot of time. Therefore, we use the sign of divisibility by 3.
- 7 + 6 + 3 + 4 + 8 = 28. The number 28 is not divisible by 3, which means 76348 is not divisible by 3 either.
- 2 + 4 + 6 + 0 + 6 = 18. The number 18 can be divided by 3 - which means that this number is divided by 3 without a remainder. Indeed, 24,606: 3 = 8,202.
In the same way, we analyze the remaining numbers:
- 1 + 1 + 2 + 8 + 9 + 4 = 25. The number 25 is not divisible by 3. Therefore, 1 128 904 is not divisible by 3.
- 5 + 4 + 0 + 8 + 1 + 3 = 21. The number 21 is divided by 3, which means that 540 813 is divided by 3. (540 813: 3 = 180 271)
Answer: 24 606 and 540 813.
When is a number divisible by 5?
However, the sign of divisibility of a number by 15 also includes not only divisibility by 3, but also the multiplicity of five.
The sign of divisibility by 5 is as follows: a number is divisible by 5 if it ends in 5 or 0.
For example, you need to find multiples of 5: 11 467, 909, 670, 840 435, 67 900
The numbers 11,467 and 909 are not divisible by 5.
The numbers 670, 840 435 and 67 900 end in 0 or 5, which means that they are a multiple of 5.
Solution Examples
So, now we can fully formulate the sign of divisibility by 15: the number is divided by 15 when the sum of its digits is a multiple of 3, and the last digit is either 5 or 0. It is important to note that both of these conditions must be fulfilled simultaneously. Otherwise, we get a multiple of not 15, but only 3 or 5.
The sign of divisibility of numbers by 15 is very often needed to solve control and examination tasks. For example, often in the basic level of the exam in mathematics there are problems based on an understanding of this particular topic. Let's consider some of their solutions in practice.
Task 1
Among the numbers, find those that are divided by 15.
9,085,475; 78,545; 531; 12,000; 90 952
So, for starters, we discard those numbers that obviously do not meet our criteria. These are 531 and 90 952. Despite the fact that the sum 5 + 3 + 1 = 9 is divided by 3, the number ends in one, which means it does not fit. The same goes for 90 952, which ends in 2.
9 085 475, 78 545 and 12 000 satisfy the first criterion, now we will check them for compliance with the second.
9 + 0 + 8 + 5 + 4 + 7 + 5 = 38, 38 is not divisible by 3. This means that this number is superfluous in our series.
7 + 8 + 5 + 4 + 5 = 29. 29 is not a multiple of 3, does not satisfy the conditions.
But 1 + 2 = 3, 3 is completely divided by 3, which means that this number is the answer.
Answer: 12,000
Task 2
The three-digit number C is greater than 700 and is divided by 15. Write down the smallest such number.
So, by the sign of divisibility by 15, this number should end with 5 or 0. Since the smallest possible one is needed, we take 0 - this will be the last digit.
Since the number is more than 700, the first may be the number 7 or more. Remembering that we should find the smallest value, choose 7.
In order for the number to be divided by 15, the condition 7 + x + 0 = a multiple of 3 must be fulfilled, where x is the number of tens.
So 7 + x + 0 = 9
X = 9 -7
X = 2
The number 720 is the desired one.
Answer: 720
Task 3.
Cross out any three digits from 3426578 so that the resulting number is a multiple of 15.
Firstly, the desired number must end with the number 5 or 0. Therefore, the last two digits - 7 and 8 must be deleted immediately.
It remains 34265.
3 + 4 + 2 + 6 + 5 = 20, 20 is not divisible by 3. The nearest multiple of 3 is 18. To get it, you need to subtract 2. Cross out the number 2.
It turns out 3465. Let's check our answer, 3465: 15 = 231.
Answer: 3465
This article examined the main signs of divisibility by 15 with examples. This material should help students with solving tasks of this type and the like, and also understand the algorithm for working with them.