What Is The Total Number Of 3-digit Numbers With Units Digit 7 And Divisible By 11?
What is the total number of 3-digit numbers with units digit 7 and divisible by 11? There are 9 such numbers. These numbers are the 3-digit numbers ending in 7 that are evenly divisible by 11. Finding these numbers involves checking divisibility rules and identifying all possible combinations.
How Do You Find All 3-digit Numbers?
To find all 3-digit numbers, you start at 100 and end at 999. This range includes every number with three digits. Each number has a hundreds place, a tens place, and a units place.
The smallest 3-digit number is 100. The largest is 999. In this range, there are 900 numbers. This is found using subtraction: 999 – 100 + 1 = 900. These numbers are used in many math problems to learn about patterns and rules.
What Is the Divisibility Rule for 11?
The divisibility rule for 11 uses alternating sums of digits. You subtract and add digit values starting from the right.
For example, the number 253. Subtract the third digit from the sum of the first two: (2 + 3) – 5 = 0. Since zero is a multiple of 11, 253 is divisible by 11. The rule helps quickly check if a number divides evenly by 11.
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Another example is 374. Check divisibility: (3 + 4) – 7 = 0. The result is zero, a multiple of 11, so it’s divisible. This method helps identify numbers ending in specific digits, like 7.
Why Focus on Units Digit 7?
The units digit 7 is our point of interest to meet the problem’s condition. Any 3-digit number we consider must end in this digit. This limits possibilities and aligns with the problem’s requirements.
Numbers ending in 7 have a specific pattern. They join others like 107, 217, or 327. Each is three digits long. Ending with 7 means these numbers are checked for different math rules, including divisibility by 11.
This condition significantly reduces the number of cases to examine. It simplifies the analysis process when determining if these numbers meet other conditions, such as divisibility rules.
How Many Such Numbers Are Divisible by 11?
There are nine numbers between 107 and 997 that meet this criteria. They include 517, 627, 737, 847, and 957.
To identify these, start at 107. Test divisibility with the alternating sum rule for 11. Add and subtract digit values. The process helps confirm divisibility. Adjust by steps of 110, since repeating is unneeded.
Each value is confirmed by this rule. Step through calculations: for example, the number 517. Calculate (5 + 7) – 1 gives 11, divisible by 11. Repeat this method for each sequence number.
Can You List All 3-digit Numbers That End in 7?
Yes, 3-digit numbers end in 7 have sequences like 107, 117, 127… up to 997. Align these numbers to test rules.
Start mathematically at 107. Increment by 10 to maintain stability for the unit’s digit at 7. Every series number will align with the pattern. This benefits those exploring number sequences and divisibility checks.
- 107
- 117
- 127
- … (continue through 997)
Once listed, apply the 11 rule for focus numbers. Identify nine valid numbers, then validate mathematical claims through calculations.
How Do You Confirm the Numbers Manually?
To confirm manually, apply the alternation sum rule to each number. Start with the sum of alternating positions.
- Start with numbers like 517. Calculate: (5 + 7) – 1 = 11.
- Number 627 follows: (6 + 7) – 2 = 11.
- Repeat for others like 737, (7 + 7) – 3 = 11.
- Ensure each result equals zero or a multiple of 11.
Verify each using simple operations. Align with table or sequence values from earlier references. Accuracy improves through checking other math calculations.
How Are These Concepts Useful in Math?
These concepts teach logical thinking and problem-solving in math. Recognizing patterns strengthens overall arithmetic comprehension.
Students gain skills in divisibility, sequential patterns, and digit-based problem solving. These tasks apply through different math levels and even in real-life problems.
Involving different rules and number characteristics encourages engagement. Repeated practice with situations like specific digits or rules builds confidence and mathematical fluency.