a cube + b cube + c cube =

a cube + b cube + c cube - 3 abc = (a+b+c) (a sq.+ b sq + c sq -ab-bc-ca)
In the sums based on this identity if LHS is given for factorisation and LHS has 3 terms (instead of 4 terms as seen above) i.e. 2 terms are perfect cube and 3rd term is not a perfect cube and has coefficient D then;
(I) first we should find the cube roots of the 2 perfect cubes,
(II) and substitute them (those) in the places of a & b above,
(III) compare the 3rd non perfect cube term with c cube - 3abc
(IV) Since only like terms can be added or subtracted, therefore you will get what should be variable part in 3rd perfect cube ( if it has so) . Let m be the coefficient part of the 3rd perfect cube
(V) and problem reduces to the form m cube + Cm = D
Here C = (-3) (coefficient part of a) (coefficient part of b)
repeating, D = coefficient part of the non-perfect cube term in the question given to solve.

m3+Cm=D

STEPS:

for small values of root of m;-

1) Find factors of D

2) Take bigger factor first say q (first FACTOR should be D itself)

3) Find (q-C) till it is a perfect square say r square

4) Divide D by this factor q to get solution r i.e. the required value of m that satisfies the above equation.

If you square this root; you will get the perfect square you got in step 3)

In other words, That factor of D is root of the above equation in variable m for which: square of answer in step 4 = answer in step 3 5) Remember factors include -ve integers also. Ex. -3 is also a factor of 15. (VI) Once you get m you get c = m x variable part obtained in step (IV)

(VII) Substitute  a, b & c in the RHS of the identity at the top , to get required factors. Simplify by clubbing like terms. 

Tricks for faster calculations - Squares of 2 digit numbers

Let ab be the number. i.e. Digit at Ten's place be a and digit at Unit's place be b.
1) Calculate square of a. Put 2 dots(..) to it's right.
2) Calculate 2 x a x b. Put 1 dot and write below above number such that right most dots in both steps are one below the other.
3) Now calculate square of b and write its unit place's digit below the right most dot written in step 2). Write rest of number below units place digit of number written in step 2)
4) Make simple addition. Treat dots as ZEROS.

Ex. 89^2
1) 8 x 8 =64 ------------------- 6 4  .  .
2) 2 x 8 x 9 = 144-------------- 1 4 4  .
3) 9 x 9 = 81 ------------------        8 1
-------------------------------------------
                                                  7 9 2 1
4) Add to get required square.

New method for division

NEW METHOD OF DIVISION.

Let us learn first how to find divisibility test for a prime number? This will lead us to the new method of division.

Divide 100 , 1000 , 10000 by the prime number. Note quotients and remainders.

Select that set of quotient and remainder where the remainder and quotients are both small numbers. It should be preferably single digit.

Suppose we want to find divisibility test for 13.

(a) 100/13 = 7; remainder 9. 13 x 7 + 9 = 100.

(b) 1000/13=76; remainder 12 or 13 x 76 +12 = 1000

(c) 10000/13=769; remainder 3

Best selection will be 7 & 9. i.e. for 100 since we are getting pair of small numbers that will be easy for calculation.

Now, any natural number (dividend) can be expressed as: 1000a+100b+10c+d , where a,b,c,d are from 0 to 9.( Whole number up to 9.i.e. 0 <= a, b, c ,d <=9 )

Now 1000a+100b+10c+d = 910a+90a+91b+9b+10c+d.

(1) Since we selected the pair of quotient and remainder for 100; leave aside last 2 digits(= no. of zeros to the right of 1 in 100).

(2) 90a+9b=9(10a+b). So multiply the number left after leaving aside last 2 digits,  by 9 (remainder).

(3) Add last 2 digits [from step (1)] to this multiplication.

(4) Divide the sum by 13. Note Quotient and Remainder. If remainder is zero the number (dividend) under consideration is divisible by 13.

(5) We know 910a+91b will be divisible by 13.and answer of division will be (a) Quotient= 7x(10a+b) i.e. 7 x the number left after leaving aside last 2 digits; (b) remainder zero since 91=13 x 7. 7 is fixed part of test.

(6) Add the quotients in step (4) & (5). This is the quotient when the dividend is divided by 13.

(7) When the number is divided by 13, the remainder will be same as that obtained in step (4).

(8) Note the test can be used as another method for division. In this method,

the only  advantage is that dividend has 2 digits less than the original number.

Engineered by

Hemant Shashikumar Dikshit

Geometry rider

If in triangle ABC, seg AB is congruent to seg AC and AP , BQ & CR are angle bisectors of angles A, B & C respectlively; (NOTE = ( is congruent to) )
prove that 1) seg CQ = seg BR
2) seg AR = seg AQ
3) seg RQ is parallel to side BC
4) Triangle ARQ is similar to triangle ABC
5) Triangle RBC is congruent to triangle QCB
6) Triangle PQR is an isosceles triangle. 

Arithmetic Progression - Triplets whose squares form an A.P.

p	d1	p+md1	p+(m+n)d1	p2	(p+md1)2	[p+(m+n)d1]2	t2-t1	t3-t2
1	1	5	7	1	25	49	24	24
1	4	29	41	1	841	1681	840	840
3	2	15	21	9	225	441	216	216
5	4	145	205	25	21025	42025	21000	21000
7	2	73	103	49	5329	10609	5280	5280
7	1	35	49	49	1225	2401	1176	1176
7	2	13	17	49	169	289	120	120
7	2	17	23	49	289	529	240	240
15	2	75	105	225	5625	11025	5400	5400
17	4	137	193	289	18769	37249	18480	18480
17	9	305	431	289	93025	185761	92736	92736
17	2	25	31	289	625	961	336	336
31	3	109	151	961	11881	22801	10920	10920
31	38	1285	1817	961	1651225	3301489	1650264	1650264
35	2	85	115	1225	7225	13225	6000	6000
41	2	89	119	1681	7921	14161	6240	6240
41	2	85	113	1681	7225	12769	5544	5544
47	1	65	79	2209	4225	6241	2016	2016
49	2	91	119	2401	8281	14161	5880	5880
51	4	159	219	2601	25281	47961	22680	22680
69	2	111	141	4761	12321	19881	7560	7560
73	5	193	263	5329	37249	69169	31920	31920
85	2	125	155	7225	15625	24025	8400	8400
89	38	1381	1951	7921	1907161	3806401	1899240	1899240
89	3	149	191	7921	22201	36481	14280	14280
97	16	593	833	9409	351649	693889	342240	342240
97	1	113	127	9409	12769	16129	3360	3360
103	34	1157	1633	10609	1338649	2666689	1328040	1328040
127	14	533	743	16129	284089	552049	267960	267960
147	2	183	213	21609	33489	45369	11880	11880
161	34	1249	1759	25921	1560001	3094081	1534080	1534080
167	30	1037	1457	27889	1075369	2122849	1047480	1047480
193	8	377	497	37249	142129	247009	104880	104880
217	4	293	353	47089	85849	124609	38760	38760
223	2	257	287	49729	66049	82369	16320	16320
223	26	925	1289	49729	855625	1661521	805896	805896
271	22	821	1129	73441	674041	1274641	600600	600600
281	26	1009	1399	78961	1018081	1957201	939120	939120
311	18	725	977	96721	525625	954529	428904	428904
329	22	901	1231	108241	811801	1515361	703560	703560
367	10	557	697	134689	310249	485809	175560	175560
383	6	485	569	146689	235225	323761	88536	88536
391	2	421	449	152881	177241	201601	24360	24360
401	14	709	919	160801	502681	844561	341880	341880
449	2	481	511	201601	231361	261121	29760	29760