How many three-digit integers are multiples of both 4 and 6 and have a unit digit of 2?
to be multiples of 4 and 6, they must be multiples of 12, the LCM
The smallest 3-digit multiple of 12 is
108 , then 120 , 132 .. .996
consider this to be an arithmetic sequence
a = 108, d = 12 , n = ? -- the number of terms
t(n) = a + (n-1)d
996 = 108 + 12(n-1)
888 = 12n - 24
12n = 912
n = 76
There are 76 of them
Since 12n = 12(5m+1)= 60m +12, our goal is to now count integers m such that 100 less than or equal to 60m +12<1000. Subtracing 12 from each part of the inequality, we get 88 lees than or equal to 60m<988. Dividing by 60, we get 2 less than or equal to m less than or equal to 16, so there are 16-1=15 values for the integer m that give us the three-digit multiples of 4 and 6 that have units digits of 2.
To find the number of three-digit integers that are multiples of both 4 and 6 and have a unit digit of 2, we need to find the common multiples of 4 and 6.
Step 1: Find the common multiples of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, ...
From the lists above, we can see that the common multiples of 4 and 6 are: 12, 24, 36, ...
Step 2: Identify the three-digit multiples of 4 and 6.
To find the three-digit multiples, we need to compare the common multiples to the range of three-digit integers (100 - 999).
The three-digit multiples of 4 and 6 are:
Multiples of 4: 108, 124, 132, 148, 164, 172, 188, 204, 212, ..., 980, 992
Multiples of 6: 108, 114, 120, 126, 132, ..., 996
Step 3: Identify the multiples with a unit digit of 2.
We can see from the lists above that only the multiples of 6 have a unit digit of 2. Therefore, the three-digit multiples of 4 and 6 with a unit digit of 2 are: 132, 192, 252, 312, 372, ..., 972.
Step 4: Count the number of three-digit multiples with a unit digit of 2.
To count the number of three-digit multiples with a unit digit of 2, we need to count the terms in the list 132, 192, 252, 312, 372, ..., 972.
Counting the terms, we can see that there are 9 three-digit multiples of both 4 and 6 with a unit digit of 2.
To find the number of three-digit integers that satisfy the given conditions, we need to break down the problem into several steps:
Step 1: Determine all the multiples of 4.
A multiple of 4 is a number that can be divided evenly by 4. The first multiple of 4 greater than or equal to 100 is 100 itself. To find the last multiple of 4 less than or equal to 999, divide 999 by 4 and round it down to the nearest whole number.
999 ÷ 4 ≈ 249.75 (round down to 249)
So, there are 249 multiples of 4 between 100 and 999.
Step 2: Determine all the multiples of 6.
A multiple of 6 is a number that can be divided evenly by 6. The first multiple of 6 greater than or equal to 100 is 102. To find the last multiple of 6 less than or equal to 999, divide 999 by 6 and round it down to the nearest whole number.
999 ÷ 6 ≈ 166.5 (round down to 166)
So, there are 166 multiples of 6 between 100 and 999.
Step 3: Determine the common multiples of 4 and 6.
To find the numbers that are multiples of both 4 and 6, we need to find the common multiples of these two numbers. The least common multiple (LCM) of 4 and 6 is 12. To find the number of common multiples between 100 and 999, divide the last multiple of 12 less than or equal to 999 by 12.
999 ÷ 12 = 83.25 (round down to 83)
So, there are 83 common multiples of 4 and 6 between 100 and 999.
Step 4: Determine the multiples of 4, 6, and have a unit digit of 2.
To have a unit digit of 2, the hundreds and tens digits can be any number, but the unit digit must be fixed as 2. Therefore, out of the common multiples calculated in step 3, only the ones with a unit digit of 2 will satisfy the given conditions.
The common multiples of 4 and 6 will have a repeating pattern of unit digits: 2, 8, 4, 0, 6, 2, 8, 4, 0, 6, ...
Thus, every 5th common multiple will have a unit digit of 2.
83 ÷ 5 ≈ 16.6
After rounding down to the nearest whole number, there are 16 common multiples of 4 and 6 that have a unit digit of 2 between 100 and 999.
Therefore, there are 16 three-digit integers that are multiples of both 4 and 6 and have a unit digit of 2.