Question

There are m multiples of 6 in range [0, 100] and n multiples of 6 in [-6, 35]. Find out the value of X, if X = m - n. Where X, m, and n are positive integer

Answer 1

given :-

• there are m multiple of 6 in range of [0, 100] and n multiple of 6 in [- 6, 35].
• m and n are positive integers.

to find :-

• what is the value of x, if x = m - n.

solution :-

• ${\small{\bold{\purple{\underline{\bigstar\: in\: case\: of\: m\: multiple\: :-}}}}}$

as we know that,

$\clubsuit$nth term of an ap formula :

$\begin{gathered}\mapsto \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}\\\end{gathered}$

where,

• $\sf a_n$ = nth term of an ap
• a = first term of an ap
• n = number of terms
• d = common difference

where,

• $\sf a_m$ = mth term of an ap
• a = first term of an ap
• m = number of terms
• d = common difference

according to the question by using the formula we get,

$\implies \sf 17 =\: m$

$\implies \sf\bold{\green{m =\: 17}}$

${\small{\bold{\purple{\underline{\bigstar\: in\: case\: of\: n\: multiple\: :-}}}}}$

given :

• first term (a) = - 6
• nth term of an ap $\sf a_n$ = 30
• common difference (d) = 6

so, for nth term we know that :

$\begin{gathered}\mapsto \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}\\\end{gathered}$

where,

• $\sf a_n$= nth term of an ap
• a = first term of an ap
• n = number of terms
• d = common difference

according to the question by using the formula we get,

$\implies \sf 30 =\: - 6 + (n - 1)6$

$\implies \sf 30 =\: - 6 + 6n - 6$

$\implies \sf 30 =\: - 6 - 6 + 6n$

$\implies \sf 30 =\: - 12 + 6n$

$\implies \sf 30 + 12 =\: 6n$

$\implies \sf 42 =\: 6n$

$\implies \sf \dfrac{\cancel{42}}{\cancel{6}} =\: n$

$\implies \sf \dfrac{7}{1} =\: n$

$\implies \sf 7 =\: n$

$\implies \sf \bold{\green{n =\: 7}}$

hence, the value of x will be :

we have :

$\bullet\: \: \sf\bold{m =\: 17}$

$\bullet\: \: \sf\bold{n =\: 7}$

then, the value of x is :

$\longrightarrow \sf x =\: m - n$

$\longrightarrow \sf x =\: 17 - 7$

$\longrightarrow \sf\bold{\red{x =\: 10}}$

$\therefore$ the value of x is 10.

Answer 2

Answer:

Answer:

Given :-

There are m multiple of 6 in range of [0, 100] and n multiple of 6 in [- 6, 35].

m and n are positive integers.

To Find :-

What is the value of X, if X = m - n.

Solution :-

${\small{\bold{\purple{\underline{\bigstar\: In\: case\: of\: m\: multiple\: :-}}}}}$

Given :

First term (a) = 0

mth term of an AP ($\sf a_m$) = 96

Common difference (d) = 6

As we know that,

$\clubsuit$ nth term of an AP Formula :

$\mapsto \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}$

where,

$\sf a_n$ = nth term of an AP

a = First term of an AP

n = Number of terms

d = Common difference

Similarly,

$\clubsuit$ mth term of an AP Formula :

$\mapsto \sf\boxed{\bold{\pink{a_m =\: a + (m - 1)d}}}$

where,

$\sf a_m$ = mth term of an AP

a = First term of an AP

m = Number of terms

d = Common difference

According to the question by using the formula we get,

$\implies \sf 96 =\: 0 + (m - 1)6$

$\implies \sf 96 =\: 0 + 6m - 6$

$\implies \sf 96 =\: 6m - 6$

$\implies \sf 96 + 6 =\: 6m$

$\implies \sf 102 =\: 6m$

$\implies \sf \dfrac{\cancel{102}}{\cancel{6}} =\: m$

$\implies \sf \dfrac{17}{1} =\: m$

$\implies \sf 17 =\: m$

$\implies \sf\bold{\green{m =\: 17}}$

${\small{\bold{\purple{\underline{\bigstar\: In\: case\: of\: n\: multiple\: :-}}}}}$

Given :

First term (a) = - 6

nth term of an AP ($\sf a_n$) = 30

Common difference (d) = 6

So, for nth term we know that :

$\mapsto \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}$

where,

$\sf a_n$ = nth term of an AP

a = First term of an AP

n = Number of terms

d = Common difference

According to the question by using the formula we get,

$\implies \sf 30 =\: - 6 + (n - 1)6$

$\implies \sf 30 =\: - 6 + 6n - 6$

$\implies \sf 30 =\: - 6 - 6 + 6n$

$\implies \sf 30 =\: - 12 + 6n$

$\implies \sf 30 + 12 =\: 6n$

$\implies \sf 42 =\: 6n$

$\implies \sf \dfrac{\cancel{42}}{\cancel{6}} =\: n$

$\implies \sf \dfrac{7}{1} =\: n$

$\implies \sf 7 =\: n$

$\implies \sf \bold{\green{n =\: 7}}$

Hence, the value of X will be :

We have :

$\bullet\: \: \sf\bold{m =\: 17}$

$\bullet\: \: \sf\bold{n =\: 7}$

Then, the value of X is :

$\longrightarrow \sf X =\: m - n$

$\longrightarrow \sf X =\: 17 - 7$

$\longrightarrow \sf\bold{\red{X =\: 10}}$

$\therefore$ The value of X is 10.