Determining the remainder...Next

Yesterday we had two problems:
P1. What is remainder when 8^102 is divided by 7?
P2. What is remainder when 9^55 is divided by 4?

Answers are:
P1. First discover the cycle for powers of 8 as divided by 7.
So 8^1 divided by 7 gives remainder 1.
This implies all powers of 8 shall yield remainder when divided by 7.
So answer is 1.

P2. 9^1 by 4 gives remainder 1.
This is similar to P1.
So answer is 1.

Let us have few more problems:

P3. What is the remainder when 6^127 is divided by 13?
P4. What remainder is generated when you divide 5^125 by 124?
P5. Can you find out what remainder is there in the division of 3^1000 by 7?

Answers and Solution in next posting.

Squaring Numbers Close to 50

Take 47. This is deficit 3 from 50.
Less 3 from 25 to get 22 and to its right place 3 square i.e. 09.
So we get 2209.

For 41, deficit is 9.
So 25-9 = 16 and place 81 to its right to get 1681.

For 57, add 7 to 25 to obtain 32 and place 47 to its right:
3249.

In case of 61, add 11 to 25 to have 36. 11 square is 121, use just two digits 21 and carry the extra 1 to 36 to make it 37. So square is 3721.

Sqauring Numbers Cloase to 1000

Take 978. It is at a deficit of 22 from 1000.
Square it to and you get 484, place these as right three digits of the answer (three digit because 1000 has three zeroes).
On left, reduce 978 by 22 to obtain 956. So the correct response is 956484.

For 991, deficit is 9 and its square is 81, which we use as 081 as we want three digits.
Left is 991-9 = 982. So square is 982484.

For 1022, add 22 to 1022 to have 1044 and to its right attach 484. The answer is 1022484.
For 1031, we have 1062961 as the square since 31 is added to 1031, and 31 square is 981.

Squaring: Keeping A Mirror At 25

Let me write a few squares around the number 25.

20^2 = 400
21^2 = 441
22^2 = 484
23^2 = 529
24^2 = 576
25^2 = 625
26^2 = 676
27^2 = 729
28^2 = 784
29^2 = 841
30^2 = 900

Can you see a similarity between numbers equally spaced on either side of 25.

So if you somehow memorize squares of numbers 1 to 25,
you will find that higher numbers can be squared very easily.

e.g. For 29, which is 4 above 25, go 4 below 25, i.e. to 21. 21^2 is 441, to this add 400 to get 841.

For 27, which is 2 above 25, go 2 below 25 to 23. Now 23 square is 529 to which you add 200 to get 729.

In fact if you want to know square of, say 39, which is 14 above 25, go 14 below 25, i.e. to 11. Add 1400 to 11 square (=121) to get 1521.

To square 49, which is 24 above 25, add 2400 to the square of number 24 below 25, which is 1. So 1^2 is 1 and adding 2400, we get 2401.

Ain't this great. As if there is a mirror at 25.

In fact if you want square of 59, which is 34 above 25, we add 3400 to square of number 34 below 25, which is -9. But square of -9 is 81, add this to 3400 to yield 3481.

Trying again for 73, we have 73 exceeding 25 by 48.
Now 4800 has to be added to what? Less 48 from 25 to get -23. Square of 23 we saw above is 529. We add this to 4800 to get 5329.

The Algebra Behind Squaring Numbers Near 100

The previous two posts may have seemed magical to many.

This however is plain algebra, the (a+b)^2 formula.

Let a number less than 100 be called (100 - d)
e.g. 93 is 100-7.

So squaring this, we get 100000 - 200d + d^2 or 100(100 - d - d) + d^2.
Isn't this what we have been doing, taking the square of d (7 in case of 93) and placing its square d^2 (or 49 in the example) to the right; since the remaining part of the answer is multiplied by 100, it leaves the right two digits untouched anyways.

And within the brackets, we have been reducing the number (=100 - d) by a further d.

In a similar vein, a number exceeding 100 would be written as (100 + e).

Its square is 10000 + 200e + e^2 or 100(100 + e + e) + e^2.
This you would see is what we did for numbers exceeding 100.

Squaring 100+ Numbers

Let us say you want to square 104.
This is 4 in excess of 100, square 4 to get 16, this gives you the right part of the answer.
Just add this 4 to 104 to obtain 108, which is to placed to left of 16 you got above,
voila! 10816 is the answer.

For 109, 9 is the excess on 100, so right part is 81. Left part is 9 added to 9, i.e. 118.
So the answer is 11881.

For 112, 12 is the excess or surplus on 100. Its square is 144, of which 44 form the right two digits of the answer, and 1 is carried to the left.
Left becomes 12 + 112, i.e. 124, with the carried 1 added, it is 125.
So the answer is 12544.

Similarly 103 square is found by squaring the surplus 3, i.e. 9 which we write as 09 as two digits have to be occupied.
The left is 3 + 103 = 106.
So the correct answer is 10609.

Squaring Carries On

Take a number like 94, it is 6 deficit from 100. So square 6 to get 36, this is the right part of the answer when you want to square 94. Reduce 94 by 6, and you get 88, this is the left part of the answer; so 94^2 = 8836.

So, for 95^2, the deficit is 5, so we have 95-5 giving 90 which is placed to left of 5 square, 9025.

For 91^2, deficit is 9; so 91-9 = 82 is placed to left of 9 square, i.e. 8281.

For 87^2, the deficit is 13, so 13^2 is 169, of which the two digits 69 are written and 1 is carried to left. Meanwhile, the left is 87-13 = 74, add the carried 1, so we get the answer 7569.

Trying 70^2, the deficit is 30, so 30^2 is 900 of which the two zeroes are retained and 9 carried to left. Alnogside, the left is 70-30=40 plus the carried 9, so the answer is 4900.