To find the smallest integer whose first n multiples all contain a 3
153×1, 153×2
1153×1, 1153×2, 1153×3
1183×1, 1183×2, 1183×3, 1183×4
3465×1, 3465×2, 3465×3, 3465×4, 3465×5
34566×1, 34566×2, 34566×3, 34566×4, 34566×5, 34566×6
53465×1, 53465×2, 53465×3, 53465×4, 53465×5, 53465×6, 53465×7
7673×1, 7673×2, 7673×3, 7673×4, 7673×5, 7673×6, 7673×7, 7673×8
Can you find an integer whose first 9 multiples all contain a 3?
What about the smallest integer whose first 10 multiples all contain a 3?
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Paul’s solutions:
first 9 multiples of 65913
first 9 multiples of 76923
first 10 multiples of 65913
first 10 multiples of 76923
first 11 multiples of 65913
first 11 multiples of 76923
first 12 multiples of 76923
first 13 multiples of 232767
first 13 multiples of 257673
first 14 multiples of 232767
first 14 multiples of 257673
first 15 multiples of 232767
first 15 multiples of 257673
first 16 multiples of 232767
first 17 multiples of 232767
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first 18 multiples of 7692391
Can you find an integer whose first 19 multiples all contain a 3?
Can you find an integer whose first 20 multiples all contain a 3?
first 21 multiples of 2307767
first 22 multiples of 3076923
first 23 multiples of 6923313
Here are some with n in range 9 to 17
65913, {65913,131826,197739,263652,329565,395478,461391,527304,593217}
76923, {76923,153846,230769,307692,384615,461538,538461,615384,692307}
65913, {65913,131826,197739,263652,329565,395478,461391,527304,593217,659130}
76923, {76923,153846,230769,307692,384615,461538,538461,615384,692307,769230}
65913, {65913,131826,197739,263652,329565,395478,461391,527304,593217,659130,725043}
76923, {76923,153846,230769,307692,384615,461538,538461,615384,692307,769230,846153}
76923, {76923,153846,230769,307692,384615,461538,538461,615384,692307,769230,846153,923076}
232767, {232767,465534,698301,931068,1163835,1396602,1629369,1862136,2094903,2327670,2560437,2793204,3025971}
257673, {257673,515346,773019,1030692,1288365,1546038,1803711,2061384,2319057,2576730,2834403,3092076,3349749}
232767, {232767,465534,698301,931068,1163835,1396602,1629369,1862136,2094903,2327670,2560437,2793204,3025971,3258738}
257673, {257673,515346,773019,1030692,1288365,1546038,1803711,2061384,2319057,2576730,2834403,3092076,3349749,3607422}
232767, {232767,465534,698301,931068,1163835,1396602,1629369,1862136,2094903,2327670,2560437,2793204,3025971,3258738,3491505}
257673, {257673,515346,773019,1030692,1288365,1546038,1803711,2061384,2319057,2576730,2834403,3092076,3349749,3607422,3865095}
232767, {232767,465534,698301,931068,1163835,1396602,1629369,1862136,2094903,2327670,2560437,2793204,3025971,3258738,3491505,3724272}
232767, {232767,465534,698301,931068,1163835,1396602,1629369,1862136,2094903,2327670,2560437,2793204,3025971,3258738,3491505,3724272,3957039}
Paul.
Nice! I posted your solutions
I posted n in range 18 to 23.
Can you find an integer for n = 19, 20?
Yes I did them just after those, but didn’t post them.
19 and 20
2307767,
3076923.
2307767.
Paul.