## smallest integer whose first n multiples all contain a 3

To find the smallest integer whose first n multiples all contain a 3

$\{153, \; 306 \}$
153×1,   153×2

$\{1153, \; 2306, \; 3459 \}$
1153×1,   1153×2,   1153×3

$\{1183, \; 2366, \; 3549, \; 4732 \}$
1183×1,   1183×2,   1183×3,   1183×4

$\{3465, \; 6930, \; 10395, \; 13860, \; 17325 \}$
3465×1,   3465×2,   3465×3,   3465×4,   3465×5

$\{34566, \; 69132, \; 103698, \; 138264, \; 172830, \; 207396 \}$
34566×1,   34566×2,   34566×3,   34566×4,   34566×5,   34566×6

$\{53465, \; 106930, \; 160395, \; 213860, \; 267325, \; 320790, \; 374255 \}$
53465×1,   53465×2,   53465×3,   53465×4,   53465×5,   53465×6,   53465×7

$\{7673, \; 15346, \; 23019, \; 30692, \; 38365, \; 46038, \; 53711, \; 61384 \}$
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

math grad - Interest: Number theory
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### 4 Responses to smallest integer whose first n multiples all contain a 3

1. Paul says:

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.

• benvitalis says:

• benvitalis says:

I posted n in range 18 to 23.
Can you find an integer for n = 19, 20?

2. Paul says:

Yes I did them just after those, but didn’t post them.

19 and 20

2307767,
3076923.

2307767.

Paul.