How many numbers between 400 to 700 both exclusive are divisible by 9 or 11?
Answers
Find the number of terms that are divisible by 9:
Find the smallest dividend:
Smallest number = dividend ÷ divisor (round up)
Smallest number = 400 ÷ 9 = 44.44 ≈ 45 (round up)
Smallest dividend = 45 x 9 = 405
Find the largest dividend:
Smallest number = dividend ÷ divisor (round down)
Smallest number = 700 ÷ 9 = 77.77 ≈ 77 (round down)
Smallest dividend = 77 x 9 = 693
Find the number of terms:
an = a1 + (n - 1)d
693 = 405 + (n - 1)9
693 = 405 + 9n - 9
693 = 9n + 396
9n = 297
n = 33
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Find the number of terms that are divisible by 11:
Find the smallest dividend:
Smallest number = dividend ÷ divisor (round up)
Smallest number = 400 ÷ 11 = 36.36 ≈ 37 (round up)
Smallest dividend = 37 x 11 = 407
Find the largest dividend:
Smallest number = dividend ÷ divisor (round down)
Smallest number = 700 ÷ 11 = 63.63 ≈ 63 (round down)
Smallest dividend = 63 x 11 = 693
Find the number of terms:
an = a1 + (n - 1)d
693 = 407 + (n - 1)11
693 = 407 + 11n - 11
693 = 11n + 396
11n = 297
n = 27
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Find the duplicates :
(Some numbers that are divisible by 9 are also divisible by 11)
Divisor = 9 x 11 = 99
Find the smallest dividend:
Smallest number = dividend ÷ divisor (round up)
Smallest number = 400 ÷ 99 = 4.04 ≈ 5 (round up)
Smallest dividend = 99 x 5 = 495
Find the largest dividend:
Smallest number = dividend ÷ divisor (round down)
Smallest number = 700 ÷ 99 = 7.07 ≈ 7 (round down)
Smallest dividend = 7 x 99 = 693
Find the number of terms:
an = a1 + (n - 1)d
693 = 495 + (n - 1)99
693 = 495 + 99n - 99
693 = 396 + 99n
99n = 297
n = 3
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Find the total number of numbers that can be divisible by 9 and 11 between 400 and 700:
33 + 27 - 3 = 57
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Answer: There are 57 numbers
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Thank you for asking this question:
We will assume as the least no divisible by 9 is 405
And the most is 693
So AP will be formed
405,414,423..........................693
The formula for nth term is given by:
an=a+(n-1)d
So, 693=405+(n-1)9
288=(n-1)9
32=n-1
So, n=33
If there is any confusion please leave a comment below.