General form???

Number Theory Level 1

If the perfect number 496 496 is expressed in the form 2 k − 1 ( 2 k − 1 ) 2^{k-1}(2^{k}-1) and the perfect number 8128 8128 is expressed in the form 2 m − 1 ( 2 m − 1 ) 2^{m-1}(2^{m}-1) .Find the value of k + m k+m

Hint:Try putting small primes in place of k k and m m .


The answer is 12.

Eddie The Head by Eddie The Head , 27, India

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1 solution

Ameya Salankar
Apr 19, 2014

By, trial & error, we find that 496 496 can be expressed as 2 5 − 1 × ( 2 5 − 1 ) 2^{5-1}\times(2^5-1) which is 16 × 31 16\times31
& we can also express 8128 8128 as 2 7 − 1 × ( 2 7 − 1 ) 2^{7-1}\times(2^7-1) which is 64 × 127 64\times127 .

Therefore, k = 5 k = 5 & m = 7 m = 7 .

⇒ k + m = 5 + 7 = 12 \Rightarrow k + m = 5 + 7 = \boxed{12} .

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on prime factorising 496, we get 496 = 2 4 × 31 { 2 }^{ 4 }\times 31 ....................... 2 5 − 1 × ( 2 5 − 1 ) { 2 }^{ 5-1 }\times ({ 2 }^{ 5 }-1)

that gives us k=5

similarly, on factorising 8128, we get 8128 = 2 6 × 127 { 2 }^{ 6 }\times 127 ................. 2 7 − 1 × ( 2 7 − 1 ) { 2 }^{ 7-1 }\times ({ 2 }^{ 7 }-1)

that gives us m=7

⟹ \Longrightarrow k+m= 12 \boxed{12}

Krishna Ramesh - 7 years, 1 month ago

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