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Use Euclids division lemma to show …

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Use Euclids division lemma to show that the cube of any positive integer of the form 9m or 9m+1 or 9m+8

Posted by Kamna Choudhary 4 years, 10 months ago

CBSE > Class 10 > Mathematics

1 answers

Jaani Kaur? 4 years, 10 months ago

Let "a" be any positive integer and 9 be another . As we know that every positive integer can be written in the form of 3q , 3q+1 , 3q+2 Also , a=9q+r where 0 is equal to or is lower than r(remainder) and 0 < 9....by euclid divison lemma Now , put r = 0 a=9q a^3 = (9q)^3 = 729q^3 = 9(81q^3) = 9m where m=81q^3 Put r=1 a=9q+1 a^3 =(9q+1)^3 = 729q^3 + 1 + 243q^2 +27q = 9(81q^3+27q^2+3q) +1 =9m+1 where m =81q^3+27q^2+3q Put r=2 a=9q+2 a^3=(9q+2)^3 =729q^3 +8+486q^2 +108q = 9(81q^3+54q^2+12q)+8 = 9m+8 where m=81q^3 +54q^2+12q Hence proved .

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