Useful for student appearing all types of competitive exam. Best for developing number sense. For more detail refer http://tinyurl.com/knmrx7n

3. 109 | वभायता-ओळखा भाग जातो का ?
2) 12 435
1 003
12 * 003 − 1 * 435
= 36 − 435 = −399
−399 स 1003 ने पूणD भाग जात नाह7 :हणून 4321528
सदेखील 1003 ने पूणD भाग जात नाह7.
iv) 159999200001 स 399999 ने पूणD भाग जातो का ?
= भाजक = 399999 = 4 00001
मसाव (4, 00001 ) = मसाव (4, −1 )
400001 चे गटवभाजन = 4 00001
भायाचे गटवभाजन = 1599992 00001
= 1600008 00001
1) 1600008 00001
4 00001
1600008 − 4 * 00001
= 1600004
2) 16 00004
4 00001
16 * 1 − 4 * 00004
= 16 − 16 = 0
0 स 399999 ने पूणD भाग जातो :हणून 159999200001
सदेखील 399999 ने पूणD भाग जातो .
109

4. SQUARING OF NUMBER UPTO 100
(ࢀࢁ)૛ = ൫(ࢀࢁ + ࢁ) ∗ ࢀ൯ ∗ ૚૙ + ࢁ૛
Examples
૚૚૛ = (૚૚ + ૚) ∗ ૚ / ૚૛ = ૚૛ /૚ = ૚૛૚
૚૛૛ = (૚૛ + ૛) ∗ ૚ / ૛૛ = ૚૝ / ૝ = ૚૝૝
૚ૠ૛ = (૚ૠ + ૠ) ∗ ૚ / ૠ૛ = ૛૝ / ૝ૢ = ૛ૡૢ
૚ૡ૛ = (૚ૡ + ૡ) ∗ ૚ / ૡ૛ = ૛૟ / ૟૝ = ૜૛૝
૛૛૛ = (૛૛ + ૛) ∗ ૛ / ૛૛ = ૝ૡ/ ૙૝ = ૝ૡ૝
૞૚૛ = (૞૚ + ૚) ∗ ૞ / ૚૛ = ૛૟૙ / ૙૚ = ૛૟૙૚
૞ૡ૛ = (૞ૡ + ૡ) ∗ ૞ / ૡ૛ = ૜૜૙ / ૟૝ = ૜૜૟૝
૞૟૛ = (૞૟ + ૟) ∗ ૞ / ૟૛ = ૜૚૙ / ૜૟ = ૜૚૜૟
ૢ૚૛ = (૚૙ૢഥ
+ૢഥ
૛ = ૚ത૚തതૡത૙/ ૚ ૡ = ૡ૛ૡ૚
) ∗ ૚૙ / ૢഥ
ૢૠ૛ = (૚૙૜ഥ
+૜ഥ
૛ = (ૢૠ − ૜) ∗ ૚૙/ ૢ ૙ = ૢ૝૙ૢ
) ∗ ૚૙ / ૜ഥ
ૢ૜૛ = (૚૙ૠഥ
+ ૠ) ∗ ૚૙ / ૠഥ
૛ = (ૢ૜ − ૠ) ∗ ૚૙/ ૢ ૝ = ૡ૟૝ૢ
Squaring of Number Near 25 , 50 , 75, 250 , 500 , 750 etc
(૛૞ + ࢇ)૛ = ૟૛૞ + ૞૙ࢇ+ ࢇ૛
= ૟૙૙ + ૚૙૙ࢇ
૛ + (૛૞ + ࢇ૛)
= ቀ૟ + ࢇ
૛ ቁ ∗ ૚૙૙ + (૛૞ + ࢇ૛)
∴ (૛૞ + ࢇ)૛ = ቀ૟+ ࢇ
૛ ቁ ࢘૛ ൫૛૞ + ࢇ૛൯
(૞૙ + ࢇ)૛ = ૛૞૙૙ + ૚૙૙ࢇ + ࢇ૛
= (૛૞ + ࢇ ) ∗ ૚૙૙ + ( ࢇ૛)
∴ (૞૙ + ࢇ)૛ = (૛૞ + ࢇ ) ࢘૛ ൫ ࢇ૛൯
Similarly,
(ૠ૞ + ࢇ)૛ = ቀ૞૟ + ૜ࢇ
૛ ቁ ࢘૛ ቀ ૛૞ + ࢇ૛ቁ

5. Shortly,
(૛૞ + ࢇ)૛ = ૟ ૛૞ ૛૞૛
+ ࢇ
૛ ࢇ૛
= ቀ૟ + ࢇ
૛ ቁ ࢘૛ (૛૞ + ࢇ૛)
(૞૙ + ࢇ)૛ = ૛૞ ૙૙ ૞૙૛
+ ࢇ ࢇ૛
= (૛૞ + ࢇ )࢘૛ (૙૙ + ࢇ૛)
(ૠ૞ + ࢇ)૛ = ૞૟ ૛૞ ૠ૞૛
+ ૜ࢇ
૛ ࢇ૛
= ቀ૞૟ + ૜ࢇ
૛ ቁ ࢘૛ (૛૞ + ࢇ૛)
(૚૙૙ + ࢇ)૛ = ૚૙૙ ૙૙ ૚૙૙૛
+ ૛ࢇ ࢇ૛
= (૚૙૙ + ૛ࢇ) ࢘૛ (૙૙ + ࢇ૛)
Similarly,
(૛૞૙ + ࢇ)૛ = ૟૛ ૞૙૙ ૛૞૙૛
+ ࢇ
૛ ࢇ૛
= ቀ૟૛ + ࢇ
૛ ቁ ࢘૜ (૞૙૙ + ࢇ૛)
(૞૙૙ + ࢇ)૛ = ૛૞૙ ૙૙૙ ૞૙૙૛
+ ࢇ ࢇ૛
= (૛૞૙ + ࢇ ) ࢘૜ ( ࢇ૛)
(ૠ૞૙ + ࢇ)૛ = ૞૟૛ ૞૙૙ ૠ૞૙૛
+ ૜ࢇ
૛ ࢇ૛
= ቀ૞૟૛ + ૜ࢇ
૛ ቁ ࢘૜ (૛૞ + ࢇ૛)
(૚૙૙૙ + ࢇ)૛ = ૚૙૙૙ ૙૙૙ ૚૙૙૙૛
+ ૛ࢇ ࢇ૛
= (૚૙૙૙ + ૛ࢇ) ࢘૜ (૙૙૙ + ࢇ૛)

6. ૛ૠ૛ = ቀ૟ + ૛
૛ቁ / ൫૛૞ + ૛૛൯ = ૠ /૛ૢ = ૠ૛ૢ
૛ૡ૛ = ቀ૟ + ૜
૛
ቁ / (૛૞ + ૜૛) = ૠ. ૞ / ૜૝ = ૠ/ ૞૙ + ૜૝ = ૠૡ૝
૛ૢ૛ = ൬૟ +
૝
૛
൰ / (૛૞ + ૝૛) = ૡ / ૝૚ = ૡ૝૚
૞૛૛ = (૛૞ + ૛) / ൫૙૙ + ૛૛൯ = ૛ૠ /૙૝ = ૛ૠ૙૝
૞૜૛ = (૛૞ + ૜) / ൫૙૙ + ૜૛൯ = ૛ૡ /૙ૢ = ૛ૡ૙ૢ
૞૟૛ = (૛૞ + ૟) / ൫૙૙ + ૟૛൯ = ૜૚ /૜૟ = ૜૚૜૟
૟૚૛ = (૛૞ + ૚૚) / ൫૙૙ + ૚૚૛൯ = ૜૟ / ૚૛૚ = ૜ૠ૛૚
૟૛૛ = (૛૞ + ૚૛) / ൫૙૙ + ૚૛૛൯ = ૜ૠ / ૚૝૝ = ૜ૡ૝૝
ૠ૟૛ = ቀ૞૟ + ૜
૛
ቁ / (૛૞ + ૚૛) = ૞ૠ. ૞ / ૛૟ = ૞ૠ/ ૞૙ + ૛૝ = ૞ૠૠ૟
૛૞૛૛ = ቀ૟૛ + ૛
૛
ቁ / (૞૙૙ + ૛૛) = ૟૜ / ૞૙૝ = ૟૜૞૙૝
૛૞ૡ૛ = ቀ૟૛ + ૡ
૛
ቁ / (૞૙૙ + ૡ૛) = ૟૟ / ૞૟૝ = ૟૟૞૟૝
૝ૢૡ૛ = ૞૙૛ഥ૛ = (૛૞૙ − ૛)/ ቀ૙૙૙ + ૛ഥ૛ቁ = ૛૝ૡ / ૙૙૝ = ૛૝ૡ૙૙૝
૞૙૝૛ = (૛૞૙ + ૝)/ (૙૙૙ + ૝૛) = ૛૞૝ / ૙૚૟ = ૛૞૝૙૚૟
૝૜૛૛ = ૞૟തതതૡത૛ = (૛૞૙ − ૟ૡ)/ ቀ૙૙૙ + ૟തതതૡത૛ቁ = ૚ૡ૛ / ૝૟૛૝ = ૚ૡ૟૟૛૝
ૢૢૡ૛ = ૚૙૙૛ഥ૛ = (૚૙૙૙ − ૝)/ ቀ૙૙૙ + ૛ഥ૛ቁ = ૢૢ૟ / ૙૙૝ = ૢૢ૟૙૙૝
ૢૢૢૡૢ૛ = (ૢૢૢૡૢ − ૚૚)/ ቀ૙૙૙૙૙ + ૚തതത૚ത૛ቁ = ૢૢૢૠૡ / ૙૙૚૛૚ = ૢૢૢૠૡ૙૙૚૛૚
Examples 1) (ࢇ࢞૛ + ࢈࢞ + ࢉ)૜ = ?
= ۺ܍ܜ ࢖ = (ࢇ࢞૛ + ࢈࢞ + ࢉ) , ࢚ࢎࢋ࢔ ࢖૜ = ࢖૛ × ࢖
By Duplex Squaring Method
࢖૛ = (ࢇ࢞૛)૛ + ૛ × (ࢇ࢞૛) × (࢈࢞) + [૛ × (ࢇ࢞૛)(ࢉ) + (࢈࢞)૛] + ૛ × (࢈࢞) × (ࢉ) + ࢉ
∴ ࢖૛ = (ࢇ૛)࢞૝ + (૛ࢇ࢈)࢞૜ + (૛ࢇࢉ + ࢈૛)࢞૛ + (૛࢈ࢉ)࢞ + ࢉ૛
∴ ࢖૜ = (ࢇ૛)࢞૝ + (૛ࢇ࢈)࢞૜ + (૛ࢇࢉ + ࢈૛)࢞૛ + (૛࢈ࢉ)࢞ + ࢉ૛
× (ࢇ࢞૛ + ࢈࢞ + ࢉ)
(ࢇ૜)࢞૟ + (૜ࢇ૛࢈)࢞૞ + (૜ࢇ࢈૛ + ૜ࢇ૛ࢉ)࢞૝ + (࢈૜ + ૟ࢇ࢈ࢉ)࢞૜ + (૜࢈૛ࢉ + ૜ࢇࢉ૛)࢞૛ + (૜࢈ࢉ૛)࢞ + ࢉ૜