Real Numbers Test - 5

4025 = 5 x 5 x 161

which is a product of prime numbers

We see that 675 > 595

675 = (595x1) + 80

595 = (80x7) + 35

80 = (35x2) + 10

35 = (10x3) + 5

10 = (5x2) + 0

The divisor in the step where the remainder is 0, is 5.

Hence the HCF is 5.

24=2 x 2 x 2 x 3

36=2 x 2 x 3 x 3

90=2 x 3 x 3 x 5

So the HCF is 2 x 3 = 6

An odd positive integer is of the form 2q+ 1 for any integer q .

Let x = 2m +1 and y = 2n +1

x² + y² = (2m + 1)² + (2n +1 )²

x² + y² = 4(m² + n²) +4(m+n) +2

x² + y² = 4{m² + n² + m + n} + 2

x² + y² is even but leaves a remainder 2, when divided by 4.

306 is a multiple of 102, so the LCM of 102, 306 and 657 is the same as the LCM of 306 and 657 which is 22338.

65 = 13 × 5 and 117 = 13 × 9

So, HCF (65, 117) = 13

Because HCF of 65 and 117 is 13,which is expressible in the form 65 m - 117

Thus we get 65 m - 117 = 13

⇒ 65 m = 130^⇒ m = 2

We see that 395 > 235

395 = (235×1) + 160

235 = (160×1) + 75

160 = (75×2) + 10

75 = (10×7) + 5

10 = (5×2) + 0

The divisor in the step where the remainder is 0, is 5.

Hence, the HCF is 5.

The reciprocal of an irrational number is an irrational number.