Ex. 21 simplify:
658x 658 x 658-328 x 328 x 328
658 x 658 + 658 x 328 + 328 x 328 =?
Sol. Given exp. (658)ᶟ - (328)ᶟ
(658)²+ (658x 328+ (328)²
= (aᶟ - bᶟ)
(a² + ab+ b²)’ where a = 658 and b = 328
= (a-b) = (658 - 328) = 330.
A = 330
Ex. 22. Simplify:
(893 + 786)² - (893 – 786)²
(893 x 786) =?
Sol Given exp. = (a+b)² - (a – b)
ab , where a = 893 and b = 786
=4 ab
ab
=4
Ex 23. What is the unit digit in the product (684 x 759 x 413 x 676)?
Sol. Unit digit in the given product
= unit digit in the product (4x 9 x3 x6) = 8.
Ex. 24. What is the unit digit in the product (3547)¹⁵³ x (251)⁷²?
Sol. Required digit = unit digit in (7¹⁵³ x 1⁷² = 1.
Now, 7⁴ gives unit digit 1 and 1⁷² = 1
(7¹⁵³ x 1⁷²) =[(7⁴)³⁸ x 7 x 1 ]
Required unit digit = (1x7x1) = 7.
Ex. 25. What is the unit digit in {(264)¹⁰² +264)¹⁰³}?
Sol. (264)¹⁰² + (264)¹⁰³+ (264)¹⁰² {1+264} = (264)¹⁰²+ 265.
Required unit digit = unit digit in [(4)¹⁰² x 5]
= unit digit in [(4)²⁵ + 4² x 5 ]
= unit digit in (6x 6x 5) =0.
Ex. 26. Find the total number of prime factors in the product {(4)¹¹ x 7⁵x (11)²}.
Sol. {(4)¹¹x 7⁵ x (11)²= (2 x2)¹¹ x 7⁵+ (11)² =(2²)¹¹x 7⁵ +(11)²
=(2²² x 7⁵ x11²)
Required number of factors = (22 + 5 + 2) =29
Ex27. Find the remainder when 2ᶟ¹ is divided by 5.
Sol. 2ᶟ¹=(2¹⁰ x 2¹⁰x 2¹⁰) x 2 = ( 2¹⁰)ᶟ x 2 = (1024)ᶟ x 2= (1024)ᶟ x 2
Unit digit in 2ᶟ¹ = unit digit in {(1024)ᶟ x 2} = (4x 2)=8
Now, 8 when divided by 5 gives 3 as remainder
2ᶟ¹ when divided by 5 gives remainder = 3.
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