Question

Remainder when (45 power 73+50 power 73) when divided by 19

Answer 1

Given :  45⁷³  +  50⁷³ divided by  19

To Find : Remainder

Solution:

45⁷³  +  50⁷³

= (38 + 7)⁷³  + (57 - 7)⁷³

(a + b)ⁿ = aⁿ  + ⁿC₁aⁿ⁻¹b¹  + ⁿC₂aⁿ⁻²b²  +................................+ ⁿCₙ₋₁a¹bⁿ⁻¹ + bⁿ

(38 + 7)⁷³  will have all term with

38   except  7⁷³

(57 - 7)⁷³  will have all term with

57   except  (-7)⁷³

38 & 57 both are divisible by 19

now 7⁷³ +  (-7)⁷³

= 7⁷³  -  7 ⁷³

= 0

Hence 45⁷³  +  50⁷³ is divisible by 19

So remainder = 0

45⁷³  +  50⁷³ divided by 19  , remainder = zero

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