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}.

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8 months ago

$\begin{array}{l} (i) G i v e n t h a t : 35 \in {x | x h a s e x a c t l y f o u r p o s i t i v e f a c t o r} \\ ∴ F a c t o r s o f 35 a r e 1, 5, 7, 35 \\ H e n c e, t h e s t a t e m e n t (i) i s' T r u e' . \\ (i i) G i v e n t h a t : 128 \in {y | t h e s u m o f a l l p o s i t i v e f a c t o r s o f y i s 2 y} \\ ∴ F a c t o r s o f 128 a r e 1, 2, 4, 8, 16, 32, 64 a n d 128 . \\ S u m o f a l l f a c t o r s = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 \\ = 255 \neq 2 * 128 \\ H e n c e, t h e s t a t e m e n t (i i) i s' F a l s e' . \\ (i i i) G i v e n t h a t : 3 \in {x | x^{4} - 5 x^{3} + 2 x^{2} - 112 x + 6 = 0} \\ ∴ x^{4} - 5 x^{3} + 2 x^{2} - 112 x + 6 = 0 \\ N o w f o r x = 3, w e h a v e \\ {(3)}^{4} - 5 {(3)}^{3} + 2 {(3)}^{2} - 112 (3) + 6 \\ \Rightarrow 81 - 135 + 18 - 336 + 6 \Rightarrow - 366 \neq 0 \\ H e n c e, t h e s t a t e m e n t (i i i) i s' T r u e' . \\ (i v) G i v e n t h a t : 496 \notin {y | t h e s u m o f a l l p o s i t i v e f a c t o r s o f y i s 2 y} \\ ∴ T h e p o s i t i v e f a c t o r s o f 496 a r e 1, 2, 4, 8, 16, 31, 62, 124, 248 a n d 496 . \\ ∴ T h e s u m o f a l l p o s i t i v e f a c t o r s = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 \\ = 992 = 2 * 496 \\ H e n c e, t h e s t a t e m e n t (i v) i s' F a l s e' . \end{array}$

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