Divide the largest 4 digit number by 49 check your solution by division algorithm

Video transcript

Let's divide 7,182 by 42. And what's different here is we're now dividing by a two-digit number, not a one-digit number, but the same idea holds. So we say, hey, how many times does 42 go into 7? Well, it doesn't really go into 7 at all, so let's add one more place value. How many times does 42 go into 71? Well, it goes into 71 one time. Just a reminder, whoever's doing the process where you say, hey, 42 goes into 71 one time. But what we're really saying, 42 goes into 7,100 100 times because we're putting this one in the hundreds place. But let's put that on the side for a little bit and focus on the process. So 1 times 42 is 42, and now we subtract. Now, you might be able to do 71 minus 42 in your head, knowing, hey, 72 minus 42 would be 30. So 71 minus 42 would be 29, but we could also do it by regrouping. To regroup, you want to subtract a 2 from a 1. You can't really do that in any traditional way. So let's take a 10 from the 70, so that it becomes a 60, and give that 10 to the ones place, and then that becomes an 11. And so 11 minus 2 is 9, and 6 minus 4 is 2. So you get 29. And we can bring down the next place value. Bring down an 8. And now, this is where the art happens when we're dividing by a multi-digit number right over here. We have to estimate how many times does 42 go into 298. And sometimes it might involve a little bit of trial and error. So you really just kind of have to eyeball it. If you make a mistake, try again. The way you know you make a mistake is, if say it goes into it 9 times, and you do 9 times 42 and you get a number larger than 298, then you overestimated. If you say it goes into it three times, you do 3 times 42, you get some number here. When you subtract, you get something larger than 42, then you also made a mistake, and you have to adjust upwards. Well, let's see if we can eyeball it. So this is roughly 40. This is roughly 300. 40 goes into 300 the same times as 4 goes into the 30, so it's going to be roughly 7. Let's see if that's right. 7 times 2 is 14. 7 times 1 is 28, plus 1 is 29. So I got pretty close. My remainder here-- notice 294 is less than 298. So I'm cool there. And my remainder is less than 42, so I'm cool as well. So now let's add another place value. Let's bring this 2 down. And here we're just asking ourselves, how many times does 42 go into 42? Well, 42 goes into 42 exactly one time. 1 times 42 is 42, and we have no remainder. So this one luckily divided exactly. 42 goes into 7,182 exactly 171 times.

Prime factors of 4, 7 and 13

4 = 2x2

7 and 13 are prime numbers.

LCM ( 4, 7, 13) = 364

We know that, the largest 4 digit number is 9999

Step 1: Divide 9999 by 364, we get

9999/364 = 171

Step 2: Subtract 171 from 9999

9999 - 171 = 9828

Step 3: Add 3 to 9828

9828 + 3 = 9831

Therefore 9831 is the number.

Find the greatest number of four digits which is exactly divisible by 15,24 and 36.

Answer

Verified

Hint: Here, we will find the L.C.M. of the given three numbers. We will divide the greatest possible four digit number by their LCM. Then by subtracting the remainder from the greatest possible four digit number we get the required greatest number of four digits which is exactly divisible by the given numbers.

Complete step-by-step answer:
First of all, by using the prime factorization method, we will find the factors of the given three numbers.
Hence, prime factorization of the first number 15 is:
$15 = 3 \times 5$
Now, prime factorization of the second number 24 is:
$24 = 2 \times 2 \times 2 \times 3 = {2^3} \times 3$
And, prime factorization of the third number 36 is:
$36 = 2 \times 2 \times 3 \times 3 = {2^2} \times {3^2}$
Now, we will find the LCM of these three numbers.
Hence, we will take all the factors present in three numbers and the highest power of the common factors respectively.
Hence, L.C.M. of these three numbers $ = {2^3} \times {3^2} \times 5 = 8 \times 9 \times 5 = 360$
Therefore, the L.C.M. of 15,24 and 36 is 360
Now, we know that the greatest number of four digits is 9999.
Now, we will divide this number by the LCM of the given three numbers.
Hence, we will use the division algorithm to find the remainder.
Dividend $=$ (Quotient $\times$ Divisor) $+$ Remainder
Substituting the values, we get
9999 $=$ (27 $\times$ 360) $+$ 279
Now, clearly, when 9999 is divided by 360, we are left with the remainder 279.
Hence, we will subtract this remainder from 9999
Thus, we get, $9999 - 279 = 9720$

Therefore, the greatest number of four digits which is exactly divisible by 15,24 and 36 is 9720.
Hence, this is the required answer.

Note:
In this question, we are required to express the given numbers as a product of their prime factors in order to find their LCM. Hence, we should know that prime factors are those factors which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Now, in order to express the given numbers as a product of their prime factors, we are required to do the prime factorization of the given numbers. Now, factorization is a method of writing an original number as the product of its various factors. Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.

What is the greatest 4 digit number that is exactly divisible by 49?

Which is the 4 digit number that can be divided by all the numbers from 1 to 10?

What is the greatest four digit number that is exactly divisible by 45 as well as 30?