Squaring numbers near 50

Last week I mentioned about the YouTube video about this problem which is said to be a Math Olympiad problem. The solution offered was also to use the difference of two squares formula: 2^18 – 1 = (2^9 + 1)( 2^9 – 1).

Then 2^9 was computed by multiplying 2 nine times: 2, 4, 8, 16, 32, 64, 128, 256 and 512.

Thus (2^9 + 1)( 2^9 – 1) = 513 x 511 = (500 + 13)(500 + 11) = 500^2 + 500 (13 + 11) + 13(11)

As we have explained last week, 2^10 bytes = 1Kb = 1024 bytes. So half of that, 2^9 is 512.

Now we have 2^18 – 1 = (2^9)^2 – 1 = 512^2 – 1

We can use the technique for squaring numbers near 50, which is similar to squaring numbers near the base as discussed in last week’s post. (https://www.math-inic.com/blog/squaring-a-number-near-a-base/)

The applicable word formula here is: Whatever the excess over 50, increase 25 by that amount and set-up the square of the excess.

Using this formula we have:

            52^2 = 25 + 2 | 2^2 = 2704

            54^2 = 25 + 4 | 4^2 = 2916

            55^2 = 25 + 5 | 5^2 = 3025

            59^2 = 25 + 9 | 9^2 = 3481

            62^2 = 25 + 12 | 12^ 2 = 37 | 144 = 3844

Note that we have allotted two places for the square of the excess.

For numbers below the base, we can use the variation: Whatever the deficiency from 50, decrease 25 by that amount and set-up the square of the deficiency.

            49^2 = (50 -1)^2 = 25 – 1 | 1^2 = 2401

            47^2 = (50 – 3)^2 = 25 – 3 | 3^2 = 2209

            39^2 = (50 -11)^2 = 25 – 11 | 11^2 = 14 | 121 = 1421

We can extend this technique for squaring numbers near 500, 5000, ETC.

            509^2 = 250 + 9 | 9^2 = 259, 081

            525^2 = 250 + 25 | 25^2 = 275, 625

            498^2 = 250 – 2 | 2^2 = 248, 004

            5015^2 = 2500 + 15 | 15^2 = 2515 | 0225 = 25, 150, 225

Now for 512^2, we have 250 + 12 | 12^2 = 262,144 and 2^18 -1 = 262,143

Exercises: find the squares of the following:

1. 53
2. 57
3. 63
4. 46
5. 44
6. 36
7. 507
8. 497
9. 5021
10. 4989

Answers to last week’s exercises:

 Find the value of the ff:

1. 12^2 = 144
2. 14^2 = 196
3. 17^2 = 289
4. 103^2 = 10609
5. 106^2 = 11236
6. 112^2 = 12544
7. 1031^2 = 1,062,961
8. 98^2 = 9604
9. 993^2 = 984,049
10. 989^2 = 978, 121