Square roots by long division

Prime fac­tori­sa­tion is slow when a num­ber is large, or when its prime fac­tors are big. The long divi­sion method works for any num­ber, step by step, one digit of the answer at a time. It also tells you how far a num­ber is from the near­est per­fect square, which helps in many puz­zles.

The steps#

1. Pair the dig­its. Start­ing from the units digit, mark the dig­its in pairs, going left. If one digit is left at the front, it is a group on its own. The num­ber of groups is the num­ber of dig­its in the square root.
2. First digit. Find the biggest num­ber whose square is not more than the first group. Write it as the first digit of the answer. Take its square away from the first group.
3. Bring down the next pair beside what is left.
4. Dou­ble the answer you have so far. Write it with a blank space on its right. Find the biggest digit $d$ that can go in the blank so that the new num­ber times $d$ is not more than the num­ber you have. Write $d$ in the answer too.
5. Take away, bring down the next pair, and repeat steps 4 and 5 until no pairs are left.

If noth­ing is left at the end, the num­ber is a per­fect square.

Exam­ple 1#

Find $\sqrt{256}$ by long divi­sion.

Pairs: $2 \mid 56$. Two groups, so the answer has two dig­its.

Step 2. The biggest square not more than $2$ is $1 \times 1 = 1$. Write $1$ in the answer. Take away: $2 - 1 = 1$.

Step 3. Bring down $56$ beside the $1$. We now have $156$.

Step 4. Dou­ble the answer so far: $1 \times 2 = 2$. Now find the digit for the blank in $2\_$:

$$\begin{aligned} 21 \times 1 &= 21 \\ 22 \times 2 &= 44 \\ 23 \times 3 &= 69 \\ 24 \times 4 &= 96 \\ 25 \times 5 &= 125 \\ 26 \times 6 &= 156 \end{aligned}$$

$26 \times 6 = 156$ exactly. Write $6$ in the answer. Take away: $156 - 156 = 0$.

$$\sqrt{256} = 16$$

This matches the answer from prime fac­tori­sa­tion.

Exam­ple 2#

Find $\sqrt{1024}$.

Pairs: $10 \mid 24$.

In step 2 we dou­bled $3$ to get $6$, and $62 \times 2 = 124$ fits exactly. (A $3$ in the blank would give $63 \times 3 = 189$, which is too big.)

$$\sqrt{1024} = 32$$

Exam­ple 3#

Find $\sqrt{97344}$.

Pairs: $9 \mid 73 \mid 44$. Three groups, so the answer has three dig­its.

Step 2: dou­ble $3$ to get $6$. $61 \times 1 = 61$ fits, but $62 \times 2 = 124$ is too big. Step 3: dou­ble $31$ to get $62$. $622 \times 2 = 1244$ fits exactly.

$$\sqrt{97344} = 312$$

Exam­ple 4#

Find $\sqrt{286225}$.

Pairs: $28 \mid 62 \mid 25$.

Step 2: dou­ble $5$ to get $10$, so the trial is $10\_$. Step 3: dou­ble $53$ to get $106$, so the trial is $106\_$.

$$\sqrt{286225} = 535$$

Exam­ple 5#

Find $\sqrt{291600}$.

Pairs: $29 \mid 16 \mid 00$.

In step 3 we bring down $00$, so we have $0$. The only digit that fits is $0$. Do not for­get to write it in the answer.

$$\sqrt{291600} = 540$$

When some­thing is left over#

If some­thing is left at the end, the num­ber is not a per­fect square. What is left tells you how far it is above the per­fect square just below it.

Exam­ple 6#

Find the least num­ber that must be taken away from $6156$ to make it a per­fect square.

Pairs: $61 \mid 56$.

$72$ is left over. So take away $72$: $6156 - 72 = 6084$, and $\sqrt{6084} = 78$.

Exam­ple 7#

Find the least num­ber that must be added to $96$ to make it a per­fect square.

$9^2 = 81$ is below $96$, and $10^2 = 100$ is above it. The next per­fect square is $100$, so add $100 - 96 = 4$.

Exam­ple 8#

An army offi­cer wants to stand $16160$ sol­diers in a solid square. He finds that $31$ sol­diers are left over. How many sol­diers are in the front row?

The sol­diers in the square are $16160 - 31 = 16129$. Long divi­sion gives $\sqrt{16129} = 127$, with noth­ing left. So the front row has $127$ sol­diers.

Remem­ber. Pair the dig­its from the right. Dou­ble the answer so far before each new step. Noth­ing left over means a per­fect square.

Com­mon mis­takes#

• Pair­ing from the left. Always pair from the units digit.
• For­get­ting to dou­ble. The trial num­ber starts with twice the answer so far.
• Leav­ing out a $0$ digit in the answer when a step gives $0$, as in $\sqrt{291600} = 540$.
• Choos­ing a digit whose prod­uct is too big. The prod­uct must not be more than the num­ber you have.

Prac­tice#

1. Find $\sqrt{1089}$ by long divi­sion.
2. Find $\sqrt{4489}$.
3. Find $\sqrt{18769}$.
4. Find $\sqrt{15129}$.
5. Find $\sqrt{120409}$.
6. Find $\sqrt{106929}$.
7. Find the least num­ber that must be taken away from $258$ to make it a per­fect square.
8. Find the least num­ber that must be added to $1000$ to make it a per­fect square.
9. Find the great­est four-digit num­ber that is a per­fect square.
10. A sports coach wants to stand $238$ stu­dents in a solid square. $13$ stu­dents are left over. How many stu­dents are in the front row?

Answers#