Question

Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication:
(i) 25
(ii) 37
(iii) 54
(iv) 71
(v) 96

Answer 1

(i) Here, a = 2, b = 5
Step 1. Make 3 columns and write the values of a², 2 x a x b, and b² in these columns.

Column I	Column II	Column III
a2	2 x a x b	b2
4	20	25

Step 2. Underline the unit digit of b² (in Column III) and add its tens digit, if any, with 2 x a x b (in Column II).

Column I	Column II	Column III
a2	2 x a x b	b2
4	20 + 2	25
	22

Step 3. Underline the unit digit in Column II and add the number formed by the tens and other digits, if any, with a² in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
4 + 2	20 + 2	25
6	22

Step 4. Underline the number in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
4 + 2	20 + 2	25
6	22

Step 5. Write the underlined digits at the bottom of each column to obtain the square of the given number.
In this case, we have:
25² = 625
Using multiplication:
25
25
125
50
625
This matches with the result obtained by the column method.

(ii) Here, a = 3, b = 7
Step 1. Make 3 columns and write the values of a², 2 x a x b, and b² in these columns.

Column I	Column II	Column III
a2	2 x a x b	b2
9	42	49

Step 2. Underline the unit digit of b² (in Column III) and add its tens digit, if any, with 2 x a x b (in Column II).

Column I	Column II	Column III
a2	2 x a x b	b2
9	42 + 4	49
	46

Step 3. Underline the unit digit in Column II and add the number formed by the tens and other digits, if any, with a² in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
9 + 4	42 + 4	49
13	46

Step 4. Underline the number in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
9 + 4	42 + 4	49
13	46

Step 5. Write the underlined digits at the bottom of each column to obtain the square of the given number.
In this case, we have:
37² = 1369
Using multiplication:
37
37
259
111
1369
This matches with the result obtained using the column method.

(iii) Here, a = 5, b = 4
Step 1. Make 3 columns and write the values of a², 2 x a x b and b² in these columns.

Column I	Column II	Column III
a2	2 x a x b	b2
25	40	16

Step 2. Underline the unit digit of b² (in Column III) and add its tens digit, if any, with 2 x a x b (in Column II).

Column I	Column II	Column III
a2	2 x a x b	b2
25	40 + 1	16
	41

Step 3. Underline the unit digit in Column II and add the number formed by the tens and other digits, if any, with a² in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
25 + 4	40 + 1	16
29	41

Step 4. Underline the number in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
25 + 4	40 + 1	16
29	41

Step 5. Write the underlined digits at the bottom of each column to obtain the square of the given number.
In this case, we have:
54² = 2916
Using multiplication:
54
54
216
270
2916
This matches with the result obtained using the column method.

(iv) Here, a = 7, b = 1
Step 1. Make 3 columns and write the values of a², 2 x a x b and b² in these columns.

Column I	Column II	Column III
a2	2 x a x b	b2
49	14	1

Step 2. Underline the unit digit of b² (in Column III) and add its tens digit, if any, with 2 x a x b (in Column II).

Column I	Column II	Column III
a2	2 x a x b	b2
49	14 + 0	1
	14

Step 3. Underline the unit digit in Column II and add the number formed by the tens and other digits, if any, with a² in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
49 + 1	14 + 0	1
50	14

Step 4. Underline the number in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
49 + 1	14 + 0	1
50	14

Step 5. Write the underlined digits at the bottom of each column to obtain the square of the given number.
In this case, we have:
71² = 5041
Using multiplication:
71
71
71
497
5041
This matches with the result obtained using the column method.

(v) Here, a = 9, b = 6
Step 1. Make 3 columns and write the values of a², 2 x a x b and b² in these columns.

Column I	Column II	Column III
a2	2 x a x b	b2
81	108	36

Step 2. Underline the unit digit of b² (in Column III) and add its tens digit, if any, with 2 x a x b (in Column II).

Column I	Column II	Column III
a2	2 x a x b	b2
81	108 + 3	36
	111

Step 3. Underline the unit digit in Column II and add the number formed by the tens and other digits, if any, with a² in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
81 + 11	108 + 3	36
92	111

Step 4. Underline the number in Column I.

Column I	Column II	Column III
a2	2 x a x b	b2
81 + 11	108 + 3	36
92	111

Step 5. Write the underlined digits at the bottom of each column to obtain the square of the given number.
In this case, we have:
96² = 9216
Using multiplication:
96
96
576
864
9216
This matches with the result obtained using the column method.