1. Perfect Square: Show
A natural number x is a perfect square if there exists a natural number y such that x=y2. In other words, a natural number x is a perfect square, if it is equal to the product of a number with itself. 2. Properties of Squares Numbers: (i) A number ending in 2, 3, 7, or 8 is never a perfect square. (ii) The number of zeroes in the end of a perfect square is never odd. So, a number ending in an odd number of zeroes is never a perfect square. (iii) Squares of even numbers are always even. (iv) Squares of odd numbers are always odd. 3. General Properties of Perfect Squares: (i) For any natural number n, we have n2= (Sum of first n odd natural numbers) (ii) The square of a natural number, other than 1, is either a multiple of 3 or exceeds a multiple of 3 by 1 . (iii) The square of a natural number, other than 1, is either a multiple of 4 or exceeds a multiple of 4 by 1. (iv) There are no natural numbers p and q such that p2=2q2 4. Pythagorean Triplets: For any natural number n greater than 1, (2n, n2−1, n2+1), is a Pythagorean triplet. 5. Square roots: The square root of a given natural number n is that natural number which when multiplied by itself gives n as the product and we denote the square root of n by n. Thus, n=m⇔n=m2. 6. Finding Square Roots: (i) In order to find the square root of a perfect square, resolve it into prime factors; make pairs of similar factors and take the product of prime factors, choosing one out of every pair. (ii) For finding the square root of a decimal fraction, make the even number of decimal places by affixing a zero, if necessary; mark off periods and extract the square root; putting the decimal point in the square root as soon as the integral part is exhausted. 7. Properties of Square Roots: For positive numbers a and b, we have (i) ab=a×b (ii) ab =ab What is the smallest number by which 2916 should be divided so that the quotient is a perfect square?Hence, the smallest number by which 2916 must be multiplied to obtain a perfect cube is 2.
What is the smallest number by which 8788 must be divided so that the product is a perfect cube?So, 2 × 2 = 4 is the least number by which 8788 should be divided so that the quotient is a perfect cube.
What is the smallest number by which 6912 must be divided so that the product is a perfect cube?Given: A number 6912 . To do: To find the smallest number by which 6912 must be divided so that the number formed is a perfect cube. Therefore, we should divide 6912 by 22=4 2 2 = 4 , the smallest number to get 1728 which is a cube of 12 .
What is the smallest number by which 1600 must be divided so that product is a perfect cube also find the number whose cube is the new number?Hence, the smallest number by which 1600 must be divided to obtain a perfect cube is 52=25.
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