Question

BRAINLIEST YR ANS AWARDED ONLY WHEN U SOLVE ALL QUESTION IF U MAKE SPAM THEN I WILL ALSO MAKE SPAMS IN YR QUESTION SECTION​

Answer 1

1.

We know the least common multiple (LCM) of two or more numbers is the smallest possible number which is exactly divisible by the individual numbers.

When we add a number less than the LCM to it, we get a number which leaves the same number, what we added to LCM, as remainder on division by the individual divisors.

Here when we add 7 to the LCM of the three divisors 30, 40 and 60, the number obtained will leave 7 as remainder on dividing by these three numbers each.

Let us find out LCM of the divisors.

$\begin{tabular}{r|ccc}\sf{10}&\sf{30,}&\sf{40,}&\sf{60}\\\cline{2-4}\sf{2}&\sf{3,}&\sf{4,}&\sf{6}\\\cline{2-4}\sf{3}&\sf{3,}&\sf{2,}&\sf{3}\\\cline{2-4}&\sf{1,}&\sf{2,}&\sf{1}\\\cline{2-4}\end{tabular}$

LCM = 10 × 2 × 3 × 2 = 120.

Hence the number is 120 + 7 = 127.

2.

The dimensions of the room are 675 cm, 450 cm, 225 cm.

The length of the largest possible measuring rod, which can measure the dimensions in exact number of times, should be the HCF of the dimensions.

$\begin{tabular}{r|ccc}\sf{225}&\sf{675,}&\sf{450,}&\sf{225}\\\cline{2-4}&\sf{3,}&\sf{2,}&\sf{1}\\\cline{2-4}\end{tabular}$

HCF = 225

Hence length of the rod is 225 cm or 2 m 25 cm.

3.

The product of two numbers is equal to the product of their LCM and HCF.

Let our number be $\sf{x,}$ so,

$\longrightarrow\sf{300x=75\times1500}$

$\longrightarrow\sf{x=\dfrac{75\times1500}{300}}$

$\longrightarrow\underline{\underline{\sf{x=375}}}$

Hence the other number is 375.

4.

Assume to reach the contradiction that $\sf{x=\sqrt6+\sqrt5}$ is a rational number.

$\longrightarrow\sf{x^2=\big(\sqrt6+\sqrt5\big)^2}$

$\longrightarrow\sf{x^2=6+5+2\times\sqrt6\times\sqrt5}$

$\longrightarrow\sf{x^2=11+2\sqrt{30}}$

$\longrightarrow\sf{\sqrt{30}=\dfrac{x^2-11}{2}}$

In this equation, $\sf{x}$ is rational, so is $\sf{x^2,}$ hence the RHS. But LHS is irrational.

This contradicts our assumption and proves that $\sf{\sqrt6+\sqrt5}$ is irrational.

5.

The LCM of any two numbers is always greater than them, or equal to any of them.

The HCF of any two numbers is always lesser than them, or equal to any of them.

If LCM and HCF of two numbers are 72 and 20 respectively,

• they should lie between 20 and 72.

• their product should be equal to 20 × 72 = 1440.

The pairs of numbers satisfying these two conditions are (36, 40) and (24, 60).

But for each pair the LCM and HCF are not 72 and 20 respectively.

Hence there does not exist pairs of numbers having LCM 72 and HCF 20.