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State which of the following statements a true and which are false. Justify your answer.

(i) 35 ∈ {x | x has exactly four positive factors}.

(ii) 128 ∈ {y | the sum of all the positive factors of y is 2y}

(iii) 3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0}

(iv) 496 ∉ {y | the sum of all the positive factors of y is 2y}.

0 5 Views | Posted 4 months ago
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    Answered by

    Vishal Baghel | Contributor-Level 10

    4 months ago

    ( i ) G i v e n     t h a t : 3 5 ∈ { x | x     h a s     e x a c t l y     f o u r     p o s i t i v e     f a c t o r } ∴         F a c t o r s     o f     3 5     a r e     1 , 5 , 7 , 3 5 H e n c e ,     t h e     s t a t e m e n t ( i )     i s     ' T r u e ' . ( i i ) G i v e n     t h a t : 1 2 8 ∈ { y | t h e     s u m     o f     a l l     p o s i t i v e     f a c t o r s     o f     y     i s     2 y } ∴         F a c t o r s     o f     1 2 8     a r e     1 , 2 , 4 , 8 , 1 6 , 3 2 , 6 4     a n d     1 2 8 .             S u m     o f     a l l     f a c t o r s = 1 + 2 + 4 + 8 + 1 6 + 3 2 + 6 4 + 1 2 8                                                                                                   = 2 5 5 ≠ 2 × 1 2 8 H e n c e ,     t h e     s t a t e m e n t ( i i )     i s     ' F a l s e ' . ( i i i ) G i v e n     t h a t : 3 ∈ { x | x 4 − 5 x 3 + 2 x 2 − 1 1 2 x + 6 = 0 } ∴                             x 4 − 5 x 3 + 2 x 2 − 1 1 2 x + 6 = 0 N o w     f o r     x = 3 ,     w e     h a v e                       ( 3 ) 4 − 5 ( 3 ) 3 + 2 ( 3 ) 2 − 1 1 2 ( 3 ) + 6 ⇒               8 1 − 1 3 5 + 1 8 − 3 3 6 + 6                     ⇒ − 3 6 6 ≠ 0 H e n c e ,     t h e     s t a t e m e n t ( i i i )     i s     ' T r u e ' . ( i v ) G i v e n     t h a t : 4 9 6 ∉ { y | t h e     s u m     o f     a l l     p o s i t i v e     f a c t o r s     o f     y     i s     2 y } ∴         T h e     p o s i t i v e     f a c t o r s     o f     4 9 6     a r e     1 , 2 , 4 , 8 , 1 6 , 3 1 , 6 2 , 1 2 4 , 2 4 8     a n d     4 9 6 . ∴             T h e     s u m     o f     a l l     p o s i t i v e     f a c t o r s = 1 + 2 + 4 + 8 + 1 6 + 3 1 + 6 2 + 1 2 4 + 2 4 8 + 4 9 6                                                                                                   = 9 9 2 = 2 × 4 9 6 H e n c e ,     t h e     s t a t e m e n t ( i v )     i s     ' F a l s e ' .

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