How many numbers less than 1,000 are multiples of both 7 and 11?
- A13
- B11
- C12
- D10
Solution & Step-by-step Explanation
To find the numbers less than 1,000 that are multiples of both 7 and 11, we need to find the numbers that are multiples of the Least Common Multiple (LCM) of 7 and 11.
Since 7 and 11 are both prime numbers:
LCM(7,11)=7×11=77
Now, we need to find how many multiples of 77 are strictly less than 1,000.
We divide 1,000 by 77:
77
1000
≈12.987
The largest integer less than or equal to this quotient is 12.
The multiples are 77×1,77×2,…,77×12.
77×12=924 (which is <1000)
Thus, there are exactly 12 such numbers.
Since 7 and 11 are both prime numbers:
LCM(7,11)=7×11=77
Now, we need to find how many multiples of 77 are strictly less than 1,000.
We divide 1,000 by 77:
77
1000
≈12.987
The largest integer less than or equal to this quotient is 12.
The multiples are 77×1,77×2,…,77×12.
77×12=924 (which is <1000)
Thus, there are exactly 12 such numbers.