Q. 44.0( 4 Votes )

# Explain using algebra, the fact that the square of any natural number which is not a multiple of 3, leaves remainder 1 on division by 3.

Answer :

We know that, We can express a number which is not a multiple of 3 as ( 3m + 1) or ( 3m +2).

( 3m+1)^{2} = (3m)^{2} + 2×3m×1 + 1^{2} = 9m^{2} + 6m + 1 = 3 ( 3m^{2} + 2m) + 1 ………..(1)

…………..using, (x+y)^{2} = x^{2} + 2xy + y^{2}

( 3m+2)^{2} = (3m)^{2} + 2×3m×2 + 2^{2} = 9m^{2} + 12m + 4 = 9m^{2} + 12m + 3 +1

= 3( 3m ^{2} + 4m +1) +1 ………..(2)

…………..using, (x+y)^{2} = x^{2} + 2xy + y^{2}

Now, observe (1) and (2) carefully, clearly on division by 3 both these terms will leave remainder 1.

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Now compute the squares of these numbers in head:

10.2

Kerala Board Mathematics Part IProve that for any natural number ending in 3, its square ends in 9. What about numbers ending in 5? And numbers ending in 4?

Kerala Board Mathematics Part IGiven below is a method to calculate 37^{2}?

Check this for some more two-digit numbers.

Kerala Board Mathematics Part IIs there a general method to compute the squares of numbers like Explain it using algebra.

Kerala Board Mathematics Part IExplain using algebra, the fact that the square of any natural number which is not a multiple of 3, leaves remainder 1 on division by 3.

Kerala Board Mathematics Part ILook at this pattern

1^{2} + (4 × 2) = 3^{2}

2^{2} + (4 × 3) = 4^{2}

3^{2} + (4 × 4) = 5^{2}

i) Write the next two lines.

ii) Explain the general principle using algebra.

Kerala Board Mathematics Part IGiven below is a method to calculate 37^{2}?

Find an easy method to compute squares of number ending in 5.

Kerala Board Mathematics Part ILook at this pattern

1^{2} + (4 × 2) = 3^{2}

2^{2} + (4 × 3) = 4^{2}

3^{2} + (4 × 4) = 5^{2}

i) Write the next two lines.

ii) Explain the general principle using algebra.

Kerala Board Mathematics Part IGiven below is a method to calculate 37^{2}?

Find an easy method to compute squares of number ending in 5.

Kerala Board Mathematics Part IGiven below is a method to calculate 37^{2}?

Explain why this is correct, using algebra.

Kerala Board Mathematics Part I