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
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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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:
Substitute a=159 and d=63.
nth term must be less than 400.
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