Which of the given numbers can be divided to get a perfect square?
(a) 8820
(b) 450
(c) 500
(d) 128
(e) 28227
(f) 2156
Answer (2)
To determine which of the given numbers can be divided to get a perfect square, we need to find a factorization of these numbers and check if dividing by any of their divisors results in a perfect square. A perfect square is a number that can be expressed as the square of an integer, like 1, 4, 9, 16, etc.
Let's examine each number one by one:
8820 :
Factor 8820:
8820 = 2 2 × 3 2 × 5 × 7 × 21
Dividing 8820 by 35 ($7 \times 5 ) gives $8820 \div 35 = 252 = 6 \times 6 = 36 , which is a perfect square as 6 2 = 36 .
450 :
Factor 450:
450 = 2 × 3 2 × 5 2
Dividing 450 by 2 gives $450 \div 2 = 225 , and $225 = 15 \times 15 , which is a perfect square as 1 5 2 = 225 .
500 :
Factor 500:
500 = 2 2 × 5 3
Dividing by 20 ($4 \times 5 ) gives $500 \div 20 = 25 = 5 \times 5 , which is a perfect square as 5 2 = 25 .
128 :
Factor 128:
128 = 2 7
Dividing 128 by 8 ($2^3 ) gives $128 \div 8 = 16 , and $16 = 4 \times 4 , which is a perfect square as $4^2 = 16 .
28227 :
Without specific factorization, if prime factored, the result is 28227 = 3 2 × 3133 . This number cannot be divided by any of its factors to get a perfect square.
2156 :
Factor 2156:
2156 = 2 2 × 7 × 77 (using a basic calculator method and factorization)
Dividing 2156 by 4 ($2^2 ): $2156 \div 4 = 539 which is not a perfect square.
From analyzing these numbers, the options that result in a perfect square when divided by certain divisors are (a) 8820 , (b) 450 , (c) 500 , and (d) 128 . They can be represented by the following factorization changes executed through consistent division for validating their perfect square status through division.
The numbers that can be divided to yield a perfect square are 8820, 450, 500, and 128. Each of these numbers can be factored and divided by specific divisors to result in a perfect square. The other numbers, 28227 and 2156, do not yield perfect squares upon division.
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