91. Required digit = Unit digit in (4x 8x 7x 3) = 2.
92. Required digit = unit digit in ( 1x 2x 3x 4x 5x 6x 7x 8 x 9) = 0.
93. (9 x 6 x 4 )= 216. In order to obtain 2at the unit place, we must multiply 216 by 2 or 7.
Of the given numbers, we have7.
94. Unit digit in (3127)¹⁷³ = unit digit in (7) ¹⁷³.now 7⁴ gives unit digit 1.
(7)¹⁷³ = (7⁴)⁴³ x 7¹.thus, (7)¹⁷³, gives unit digit 7.
95. Unit digit in 7⁴ is 1.
Unit digit in 7⁶⁸ is 1.
Unit digit in 7⁷¹ is 3.
Again, every power of 6 will give unit digit 6.
Unit digit in 6⁵⁹ is 6.
Unit digit in 3⁴ is 1.
Unit digit in 3⁶⁴ is 1. Unit digit in 3⁶⁵ is 3.
Unit digit in(⁷¹ x 6⁵⁹ x 3⁶⁵) = unit digit in (3 x 6 x 3) = 4.
96. Unit digit in 7⁴ is 1. So, unit digit in 7⁹² is 1.
Unit digit in 7⁹⁵ is 3.
Unit digit in 3⁴ is 1.
Unit digit in 3⁵⁶ is 1.
Unit digit in3⁵⁸is 9.
Unit digit in (7⁹⁵ - 3⁵⁸) = (13 – 9) = 4.
97. X⁴ᶮ = (2⁴)ᶮ or (4⁴)ᶮ or (6⁴)ᶮ or (8⁴)n/
Clearly, the unit digit in each case is 6.
98. (3x 5)¹² x (2 x 7)¹⁰ x (10)²⁵ = (3x 5)¹² x (2x 7)¹⁰ x (2 x 5)²⁵
= 3¹² x 5¹² x 2¹⁰ x 7¹⁰x 2²⁵ x 5²⁵ = 2³⁵ x 3¹² x 5³⁷ x 7¹⁰.
Total number of prime factors = (35 + 12 + 37 + 10) = 94.
99. Given exp. = a² + b² + 2ab , where a =397 and b= 104.
= (a + b)²= (397 + 104)² = (500 + 1)² =(500)² + 1² + 2 x 500 x1
= (250000 + 1 + 1000 = 251001. ²
100. Given exp. = = a² + b² + 2ab, where a =186 and b= 159
= (a + b)²=(186 -159)² = (27)²
=(20 + 7)² = (20)²+ 7² + 2 x 20 x 7 =400 + 49 + 280 = 729.
