Question
upstudy study bank question image url

How many of the numbers from 10 through 83 have the sum of their digits equal to a perfect square? There are \( \square \) numbers whose digits sum to a perfect square.

Ask by Rodriquez Gough. in the United States
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

There are 12 numbers between 10 and 83 where the sum of their digits is a perfect square.

Solution

To solve the problem, we need to find how many numbers from 10 to 83 have the sum of their digits equal to a perfect square. ### Step 1: Identify the range The numbers we are considering are from 10 to 83, inclusive. ### Step 2: Calculate the sum of digits For each number \( n \) in this range, we will calculate the sum of its digits. ### Step 3: Identify perfect squares The perfect squares less than or equal to the maximum possible sum of digits (which is \( 8 + 3 = 11 \) for the number 83) are: - \( 0^2 = 0 \) - \( 1^2 = 1 \) - \( 2^2 = 4 \) - \( 3^2 = 9 \) Thus, the perfect squares we will consider are \( 0, 1, 4, 9 \). ### Step 4: Check each number We will check each number from 10 to 83, calculate the sum of its digits, and see if it matches any of the perfect squares identified. Let's calculate the sum of digits for each number in the range and count how many of them have sums that are perfect squares. ### Step 5: Count the valid numbers We will iterate through each number from 10 to 83, calculate the sum of its digits, and check if it is a perfect square. Now, let's perform the calculations. - For \( n = 10 \): \( 1 + 0 = 1 \) (perfect square) - For \( n = 11 \): \( 1 + 1 = 2 \) - For \( n = 12 \): \( 1 + 2 = 3 \) - For \( n = 13 \): \( 1 + 3 = 4 \) (perfect square) - For \( n = 14 \): \( 1 + 4 = 5 \) - For \( n = 15 \): \( 1 + 5 = 6 \) - For \( n = 16 \): \( 1 + 6 = 7 \) - For \( n = 17 \): \( 1 + 7 = 8 \) - For \( n = 18 \): \( 1 + 8 = 9 \) (perfect square) - For \( n = 19 \): \( 1 + 9 = 10 \) - For \( n = 20 \): \( 2 + 0 = 2 \) - For \( n = 21 \): \( 2 + 1 = 3 \) - For \( n = 22 \): \( 2 + 2 = 4 \) (perfect square) - For \( n = 23 \): \( 2 + 3 = 5 \) - For \( n = 24 \): \( 2 + 4 = 6 \) - For \( n = 25 \): \( 2 + 5 = 7 \) - For \( n = 26 \): \( 2 + 6 = 8 \) - For \( n = 27 \): \( 2 + 7 = 9 \) (perfect square) - For \( n = 28 \): \( 2 + 8 = 10 \) - For \( n = 29 \): \( 2 + 9 = 11 \) - For \( n = 30 \): \( 3 + 0 = 3 \) - For \( n = 31 \): \( 3 + 1 = 4 \) (perfect square) - For \( n = 32 \): \( 3 + 2 = 5 \) - For \( n = 33 \): \( 3 + 3 = 6 \) - For \( n = 34 \): \( 3 + 4 = 7 \) - For \( n = 35 \): \( 3 + 5 = 8 \) - For \( n = 36 \): \( 3 + 6 = 9 \) (perfect square) - For \( n = 37 \): \( 3 + 7 = 10 \) - For \( n = 38 \): \( 3 + 8 = 11 \) - For \( n = 39 \): \( 3 + 9 = 12 \) - For \( n = 40 \): \( 4 + 0 = 4 \) (perfect square) - For \( n = 41 \): \( 4 + 1 = 5 \) - For \( n = 42 \): \( 4 + 2 = 6 \) - For \( n = 43 \): \( 4 + 3 = 7 \) - For \( n = 44 \): \( 4 + 4 = 8 \) - For \( n = 45 \): \( 4 + 5 = 9 \) (perfect square) - For \( n = 46 \): \( 4 + 6 = 10 \) - For \( n = 47 \): \( 4 + 7 = 11 \) - For \( n = 48 \): \( 4 + 8 = 12 \) - For \( n = 49 \): \( 4 + 9 = 13 \) - For \( n = 50 \): \( 5 + 0 = 5 \) - For \( n = 51 \): \( 5 + 1 = 6 \) - For \( n = 52 \): \( 5 + 2 = 7 \) - For \( n = 53 \): \( 5 + 3 = 8 \) - For \( n = 54 \): \( 5 + 4 = 9 \) (perfect square) - For \( n = 55 \): \( 5 + 5 = 10 \) - For \( n = 56 \): \( 5 + 6 = 11 \) - For \( n = 57 \): \( 5 + 7 = 12 \) - For \( n = 58 \): \( 5 + 8 = 13 \) - For \( n = 59 \): \( 5 + 9 = 14 \) - For \( n = 60 \): \( 6 + 0 = 6 \) - For \( n = 61 \): \( 6 + 1 = 7 \) - For \( n = 62 \): \( 6 + 2 = 8 \) - For \( n = 63 \): \( 6 + 3 = 9 \) (perfect square) - For \( n = 64 \): \( 6 + 4 = 10 \)

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

The Deep Dive

To find how many numbers from 10 to 83 have the sum of their digits equal to a perfect square, we first need to identify the possible perfect squares within the range of digit sums. The relevant perfect squares less than or equal to 16 (the maximum sum of digits for 83, which is 8 + 3 = 11) are 1, 4, 9, and 16. Now, let's calculate the digit sums for numbers from 10 to 83: - For **10 to 19**: - 10: 1, 11: 2, 12: 3, 13: 4, 14: 5, 15: 6, 16: 7, 17: 8, 18: 9, 19: 10 Perfect Square sums: 4 (13), 9 (18). Total = 2. - For **20 to 29**: - 20: 2, 21: 3, 22: 4, 23: 5, 24: 6, 25: 7, 26: 8, 27: 9, 28: 10, 29: 11 Perfect Square sums: 4 (22), 9 (27). Total = 2. - For **30 to 39**: - 30: 3, 31: 4, 32: 5, 33: 6, 34: 7, 35: 8, 36: 9, 37: 10, 38: 11, 39: 12 Perfect Square sums: 4 (31), 9 (36). Total = 2. - For **40 to 49**: - 40: 4, 41: 5, 42: 6, 43: 7, 44: 8, 45: 9, 46: 10, 47: 11, 48: 12, 49: 13 Perfect Square sums: 4 (40), 9 (45). Total = 2. - For **50 to 59**: - 50: 5, 51: 6, 52: 7, 53: 8, 54: 9, 55: 10, 56: 11, 57: 12, 58: 13, 59: 14 Perfect Square sums: 4 (53), 9 (54). Total = 2. - For **60 to 69**: - 60: 6, 61: 7, 62: 8, 63: 9, 64: 10, 65: 11, 66: 12, 67: 13, 68: 14, 69: 15 Perfect Square sums: 4 (64), 9 (63). Total = 2. - For **70 to 79**: - 70: 7, 71: 8, 72: 9, 73: 10, 74: 11, 75: 12, 76: 13, 77: 14, 78: 15, 79: 16 Perfect Square sums: 9 (72), 16 (79). Total = 2. - For **80 to 83**: - 80: 8, 81: 9, 82: 10, 83: 11 Perfect Square sums: 9 (81). Total = 1. Now let's sum them up: 2 (10-19) + 2 (20-29) + 2 (30-39) + 2 (40-49) + 2 (50-59) + 2 (60-69) + 2 (70-79) + 1 (80-83) = 15. There are \( 15 \) numbers whose digits sum to a perfect square.

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy