A troops can be arranged in 6,7 and 8 rows, but not is a square form. (a) Find out factors of 8 . (b) What is the least number by which the number in troops is to multiplied so that the troops can be arranged in a square form? (c) At least how many troops should have to join to arrange troop btained in a square form?

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We are told that a certain number of troops can be arranged in 6 rows, 7 rows, or 8 rows (that is, the total number is divisible by 6, 7, and 8) but they cannot form a perfect‐square array. In parts (b) and (c) we are asked how to modify that number so that it becomes a perfect square. (Also, part (a) asks a simple question about the factors of 8.)

In many such problems the “least” number of troops satisfying the row–arrangement condition is taken. Since the number must be divisible by 6, 7, and 8, the smallest such number is the least common multiple (LCM) of 6, 7 and 8. Let’s start there.

Step 1. Find the LCM of 6, 7 and 8.

• 6 = 2 × 3  
 • 7 is prime  
 • 8 = 2³

Thus, the LCM must have 2 raised to the highest exponent (which is 3), a factor 3 and a factor 7. That is,
  LCM = 2³ × 3 × 7 = 8 × 3 × 7 = 168.

So we take the number of troops as N = 168.

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Part (a). Find out the factors of 8.

The positive factors (divisors) of 8 are the numbers that evenly divide 8. They are:
  1, 2, 4, and 8.

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Part (b). What is the least number by which the number in troops is to be multiplied so that the troops can be arranged in a square form?

We want to multiply 168 by the smallest possible whole-number factor m such that m×168 is a perfect square. (A perfect square has all prime exponents even in its prime–factorization.)

First, write 168 in its prime–factorization:
  168 = 2³ × 3¹ × 7¹

For a number to be a perfect square each prime’s exponent must be even. Here the exponents are 3, 1, and 1—all odd (or not even). To fix this, we need to supply extra factors:
 • For the 2’s: 2³ needs one more 2 (because 3 + 1 = 4, which is even).  
 • For the 3’s: 3¹ needs one more 3.  
 • For the 7’s: 7¹ needs one more 7.

Thus, the minimum m is:
  m = 2 × 3 × 7 = 42

Checking: 168 × 42 = (2³ × 3 × 7) × (2 × 3 × 7) = 2⁴ × 3² × 7², which is a perfect square, namely (2²×3×7)² = (4×3×7)² = (84)².

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Part (c). At least how many troops should have to join so that the troops can be arranged in a square form?

Here the idea is that if a few extra troops join the existing troops the total number will become a perfect square. We are not multiplying
(a) The factors of 8 are 1, 2, 4, and 8. (b) To arrange the troops in a square formation, multiply the number of troops by 42. (c) At least 42 troops need to join to arrange the total number of troops in a square formation.

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