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.
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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.