Verified answer

★ Figure refer to attachment

Observations :-

✰Circumcentre✰ :- The point where the perpendicular bisectors of the sides of a triangle meet.

➞ Perpendicular bisectors of the sides of ∆ABC intersect at O

➞ OA = OB = OC are the radius of the given circle with centre O

➞ O is the circumcentre

✰Incentre✰ :- The point where the angle bisectors of a triangle meet.

➞ Angle bisectors of ∆ABC intersect at I

➞ ID is the radius of circle with centre I

➞ I is the Incentre

✰Centroid✰ :- The point where the three medians of a triangle meet.

➞ The line segment that joined a vertex of the triangle to the midpoint of the opposite side is known as median.

➞ A triangle has always three medians

➞ AG, BG & CG of ∆ABC intersect at the point G.

➞ G is the centroid

✰Orthocentre✰ :- The point where the three altitudes of a triangle meet.

➞ The line segment from a vertex that is perpendicular to opposite side.

➞ GI ⊥ FE , EH ⊥ FG & FI ⊥ GE {Altitudes}

➞ A triangle has always three altitudes.

➞ GI, HE & FJ of ∆GEF intersect at point O

➞ O is the orthocentre

Note : The point in which three or more lines intersect each other at one spot is known as point of concurrency

Now,

• 31944

2 | 31944

2 | 15972

2 | 7986

3 | 3993

11| 1331

11 | 121

11 | 11

| 1

✦31944 = 2 × 2 × 2 × 3 × 11 × 11 × 11

*Clearly, to make it a perfect cube, it must be divided by 3

• 64827

3 | 64827

3 | 21609

3 | 7203

7 | 2401

7 | 343

7 | 49

7 | 7

| 1

✦64827 = 3 × 3 × 3 × 7 × 7 × 7 × 7

*Clearly, to make it a perfect cube, it must be divided by 7

• 200704

2 | 200704

2 | 100352

2 | 50176

2 | 25088

2 | 12544

2 | 6272

2 | 3136

2 | 1568

2 | 784

2 | 392

2 | 196

2 | 98

7 | 49

7| 7

| 1

✦200704 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2× 7 × 7

*Clearly, to make it a perfect cube, it must be divided by (7 × 7) i.e 49