Math, asked by mohittanwar5779, 1 year ago

16. How many numbers lie between
100 and 400 which when divided
by 9 leave a remainder 6, and
when divided by 21, leave
remainder 12 ?​

Answers

Answered by bhagyashreechowdhury
9

There are 4 numbers- 159, 222, 285 & 384 that lie between 100 and 400 which when divided by 9 leave a remainder 6, and when divided by 21, leave remainder 12.

Step-by-step explanation:

Required formula:

Dividend = Divisor*Quotient + Remainder

Let’s assume the quotient in case of 9 as a divisor be “x” and the quotient in case of 21 as a divisor be “y”.

So, based on the formula and the data given in the question we can write the equation as,

9x + 6  = 21y + 12

⇒ 9x  = 21y + 6

⇒  3x = 7y + 2 ……. (i)

Here we are given that, the numbers should be between 100 & 400, therefore, we can also write

21y + 12 > 100

⇒ 21y > 88

y >  4.1 == 4 …… (ii)

and,

21y + 12 < 400

⇒ 21y < 388

⇒ y < 18.47 == 19 ……. (iii)

Now from (ii) & (iii), we will put the values of y between 4 and 19 in eq. (i) in such a way that “3x” is exactly divisible by “7y+2”, therefore,

  • When y = 7

                    3x = 7*7 + 2

                   ⇒ x = 51/3 = 17

  • When y = 10    

                     3x = 7*10 + 2

                      ⇒ x = 72/3 = 24

  • When y = 13  

                   3x = 7*13 + 2

                   ⇒ x = 93/3 = 31

  • When y = 16    

                  3x = 7*16 + 2

                  ⇒ x = 114/3 = 38

Substituting all the values of x or y so received above in “[9x + 6]” or “[21y + 12]” respectively we will get the numbers between 100 and 400 which when divided by 9 leave a remainder 6, and when divided by 21, leave remainder 12 which are:

9*17 +6 = 159

9*24 + 6 = 222

9*31 + 6 = 285

9*38 + 6 = 384

Thus, the numbers are 159, 222, 285 & 384.

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Answered by FelisFelis
4

Answer:

The there are 4 terms that lie between 100 and 400.

Step-by-step explanation:

Consider the provided information.

We need to find how many number lie between 100 and 400  which when divided  by 9 leave a remainder 6, and  when divided by 21, leave  remainder 12.

First find the first number from 0 which gives remainder 6 when divided by 9 and gives remainder 12 when divided by 21.

The first number is 33 when divided by 9 and gives remainder 12 when divided by 21.

Now find the LCM of 9 and 21.

9=3×3

21=3×7

The LCM of the number 9 and 21 is 63.

The first number from 0 is 33 and the other numbers are separated by 63.

Therefore the required sequence is: 33,96,159.....

Here we can observe that between 100 and 400 first number is 159 and the common ratio is 63 so use AP formula as shown below:

a_n=a+(n-1)d

Substitute a=159 and d=63.

nth term must be less than 400.

a+(n-1)d&lt;400

159+(n-1)63&lt;400

(n-1)63&lt;241

n-1&lt;3.82

n&lt;4.82

Hence, the there are 4 terms that lie between 100 and 400.

The 4 numbers that lie between 100 and 400 which when divided by 9 leave a remainder 6, and when divided by 21, leave remainder 12  are:

159, 222, 285, 348

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