What is the least number that must be added to 1357 to get a perfect square?

If you want to find the least number which has to be added to x to get a perfect square, find the square root of x using long division.

When you find square root of x using long division, let k be the remainder where k > 0 and q be the quotient.

The square of the quotient q is q2 and

q2 < x ----(1)

The next perfect square number is (q + 1)2 and 

(q + 1)2 > x ----(2)

From (1) and (2),

q2 < x < (q + 1)2

Find the difference between x and (q + 1)2.

Say, (q + 1)2 - x = p.

p is the least number to be added to x to get a perfect square.

Thus, (x + p) is a perfect square.

Example 1 :

Find the least number which must be added to 104979 so as to get a perfect square.

Solution :

Find the square root of 104979 using long division method.

Step 1 :

Separate the digits by taking commas from right to left once in two digits.

10,49,79

When we do so, we get 10 before the first comma.

Step 2 :

Now we have to multiply a number by itself such that

the product ≤ 10

(The product must be greatest and also less than 10)

The above condition will be met by '3'.

Because 3 ⋅ 3 = 9 ≤ 10.

In the above picture, 9 is subtracted from 10 and we got the remainder 1.

Step 3 :

Now, we have to bring down 49 and quotient 3 to be multiplied by 2 as given in the picture below.

Step 4 :

Now we have to take a same number at the two places indicated by '?'.

Then, we have to find the product as shown in the picture and also the product must meet the condition as indicated.

Step 5 :

The condition said in step 4 will be met by replacing '?' with '2'.

Than we have to do the calculation as given in the picture.

Step 6 :

Now, we have to bring down 79 and quotient 32 to be multiplied by 2 as given in the picture below.

Step 7 :

In finding square root of 104979 using long division, we obtain the remainder 3 and the quotient is 324.

3242 = 104976 < 104979 ----(1)

The next perfect square number :

3252 = 105625 > 104979 ----(2)

From (1) and (2),

3242 < 104979 < 3252

Find the difference between 104979 and 3252 :

3252 - 104979 = 105625 - 104979

= 646

646 is the least number must be added to 104979 to get a perfect square.

Example 2 :

Find the least number which must be added to 1989 so as to get a perfect square.

Solution :

Find the square root of 1989 using long division method.

In finding square root of 1989 using long division, we obtain the remainder 53 and the quotient is 44.

442 = 1936 < 1989 ----(1)

The next perfect square number :

452 = 2025 > 1989 ----(2)

From (1) and (2),

442 < 1989 < 452

Find the difference between 1989 and 452 :

452 - 1989 = 2025 - 1989

= 36

36 is the least number must be added to 1989 to get a perfect square.

Example 3 :

Find the least number which must be added  to 3250 so as to get a perfect square.

Solution :

Find the square root of 3250 using long division method.

In finding square root of 3250 using long division, we obtain the remainder 1 and the quotient is 57.

572 = 3249 < 3250 ----(1)

The next perfect square number :

582 = 3364 > 3250 ----(2)

From (1) and (2),

572 < 3250 < 452

572 < 3250 < 582

Find the difference between 3250 and 582 :

582 - 3250 = 3364 - 3250

= 114

114 is the least number must be added to 3250 to get a perfect square.

Example 4 :

Find the least number which must be added to 4000 so as to get a perfect square.

Solution :

Find the square root of 4000 using long division method.

In finding square root of 4000 using long division, we obtain the remainder 31 and the quotient is 63.

Here we will define, analyze, simplify, and calculate the square root of 1357. We start off with the definition and then answer some common questions about the square root of 1357. Then, we will show you different ways of calculating the square root of 1357 with and without a computer or calculator. We have a lot of information to share, so let's get started!

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Square root of 1357 definition
The square root of 1357 in mathematical form is written with the radical sign like this √1357. We call this the square root of 1357 in radical form. The square root of 1357 is a quantity (q) that when multiplied by itself will equal 1357.

√1357 = q × q = q2

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Is 1357 a perfect square?
1357 is a perfect square if the square root of 1357 equals a whole number. As we have calculated further down on this page, the square root of 1357 is not a whole number.

1357 is not a perfect square.

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Is the square root of 1357 rational or irrational?
The square root of 1357 is a rational number if 1357 is a perfect square. It is an irrational number if it is not a perfect square. Since 1357 is not a perfect square, it is an irrational number. This means that the answer to "the square root of 1357?" will have an infinite number of decimals. The decimals will not terminate and you cannot make it into an exact fraction.

√1357 is an irrational number

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Can the square root of 1357 be simplified?
You can simplify 1357 if you can make 1357 inside the radical smaller. We call this process "to simplify a surd". The square root of 1357 cannot be simplified.

√1357 is already in its simplest radical form.

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How to calculate the square root of 1357 with a calculator
The easiest and most boring way to calculate the square root of 1357 is to use your calculator! Simply type in 1357 followed by √x to get the answer. We did that with our calculator and got the following answer with 9 decimal numbers:

√1357 ≈ 36.837480913

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How to calculate the square root of 1357 with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(1357) in a cell to get the square root of 1357. Below is the result we got with 13 decimals. We call this the square root of 1357 in decimal form.

SQRT(1357) ≈ 36.8374809127877

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What is the square root of 1357 rounded?
The square root of 1357 rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of 1357 rounded to the nearest hundredth, means that you want two digits after the decimal point. The square root of 1357 rounded to the nearest thousandth, means that you want three digits after the decimal point.

10th: √1357 ≈ 36.8

100th: √1357 ≈ 36.84

1000th: √1357 ≈ 36.837

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What is the square root of 1357 as a fraction?
Like we said above, since the square root of 1357 is an irrational number, we cannot make it into an exact fraction. However, we can make it into an approximate fraction using the square root of 1357 rounded to the nearest hundredth.

√1357
≈ 36.84/1
≈ 3684/100
≈ 36 21/25

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What is the square root of 1357 written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of 1357 is no exception. Here is the rule and the answer to "the square root of 1357 converted to a base with an exponent?":

√b = b½

√1357 = 1357½

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How to find the square root of 1357 by long division method
Here we will show you how to calculate the square root of 1357 using the long division method with one decimal place accuracy. This is the lost art of how they calculated the square root of 1357 by hand before modern technology was invented.

Step 1)
Set up 1357 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:

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Step 2)
Starting with the first set: the largest perfect square less than or equal to 13 is 9, and the square root of 9 is 3. Therefore, put 3 on top and 9 at the bottom like this:

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Step 3)
Calculate 13 minus 9 and put the difference below. Then move down the next set of numbers.

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Step 4)
Double the number in green on top: 3 × 2 = 6. Then, use 6 and the bottom number to make this problem:

6? × ? ≤ 457

The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 6. Replace the question marks in the problem with 6 to get:

66 × 6 = 396.

Now, enter 6 on top, and 396 at the bottom:

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Step 5)
Calculate 457 minus 396 and put the difference below. Then move down the next set of numbers.

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Step 6)
Double the number in green on top: 36 × 2 = 72. Then, use 72 and the bottom number to make this problem:

72? × ? ≤ 6100

The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 8. Now, enter 8 on top:

That's it! The answer is on top. The square root of 1357 with one digit decimal accuracy is 36.8. Did you notice that the last two steps repeat the previous two steps. You can add decimals by simply adding more sets of 00 and repeating the last two steps over and over.

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Square Root of a Number
Please enter another number in the box below to get the square root of the number and other detailed information like you got for 1357 on this page.

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Notes
Remember that negative times negative equals positive. Thus, the square root of 1357 does not only have the positive answer that we have explained above, but also the negative counterpart.

We often refer to perfect square roots on this page. You may want to use the list of perfect squares for reference.

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