Question

How many 3 -digit numbers are multiples of both 20 and 35?

Answer 1

The number of 3-digit numbers that are multiples of both 20 and 35 is 7. To find this, calculate the least common multiple (LCM) of 20 and 35, which is 140, then determine how many multiples of 140 fit within the 3-digit range.

Step-by-step explanation:

To find out how many 3-digit numbers are multiples of both 20 and 35, we need to determine the least common multiple (LCM) of 20 and 35 first. The LCM of two numbers is the smallest number that is a multiple of both. The prime factors of 20 are 22 × 5, and the prime factors of 35 are 5 × 7. So, the LCM of 20 and 35 is 22 × 5 × 7 = 140. Now, we must find all the multiples of 140 that are 3-digit numbers.

The smallest 3-digit multiple of 140 is 140 itself, and the largest is 980, as the next multiple, 1120, is a 4-digit number. We can find the number of multiples by dividing the largest by the LCM and then subtracting the result of dividing the smallest by the LCM, then adding 1.

Therefore, the calculation is (980 ÷ 140) - (140 ÷ 140) + 1, which gives us (7 - 1) + 1 = 7. There are 7 three-digit numbers that are multiples of both 20 and 35.

Answer 2

The number of 3-digit numbers that are multiples of both 20 and 35 is 6. This is determined by finding the LCM of 20 and 35, which is 140, and counting how many times 140 fits into the range of 3-digit numbers from 100 to 999.

Step-by-step explanation:

To find the number of 3-digit numbers that are multiples of both 20 and 35, we first need to determine the least common multiple (LCM) of these two numbers. The LCM of 20 (which is 22 × 5) and 35 (which is 5 × 7) is 22 × 5 × 7, or 140. This is the smallest number that is a multiple of both 20 and 35.

Next, we look for the range of 3-digit numbers (100 to 999) and see how many times 140 can fit into this range. The smallest 3-digit multiple of 140 is 140 × 1 = 140, which is not a 3-digit number, so the next multiple would be 140 × 2 = 280. The largest 3-digit multiple of 140 is 140 × 7 = 980.

Now we count how many multiples of 140 there are from 280 to 980. Starting at 280 (140 × 2), we get: 280, 420, 560, 700, 840, and 980. That's a total of 6 numbers.

Therefore, there are 6 three-digit numbers that are multiples of both 20 and 35.