Maud from Hymers college in England and Luke from Long field Academy, both in England, provided Alison"s an approach to counting the components of 360. Maud wrote:

To discover the determinants you require to know what components are, determinants can"t be over the number you room trying to discover the components for, so you would certainly divide 360 by a number (I would start at 1 then save going up) and also that would be 360 for this reason 360 and 1 are both factors and you would carry out this because that the rest until that repeats itself.

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Holly and also Amy indigenous Long field Academy and Sanika indigenous India used Charlie"s an approach to count how many factors 360 has. This is Holly"s work: \$360 = 2^3 imes3^2 imes5^1\$

Number of components \$=4 imes3 imes2 = 24\$

Alex and also Amy native Long field Academy and Sanika likewise used Charlie"s an approach to display that Alison and also Charlie"s other numbers have actually 24 factors. This is Alex"s work:

25725= 52\$ imes\$31\$ imes\$73

Click to watch Alex"s table

 50 30 70 1 71 7 72 49 73 343 31 70 3 71 21 72 147 73 1029 51 30 70 5 71 35 72 245 73 1715 31 70 15 71 105 72 735 73 5145 52 30 70 25 71 175 72 1225 73 8575 31 70 75 71 525 72 3675 73 25725

So there will be 3 rows in the an initial column which enhance the 5. Additionally in the second column there will certainly be 2 rows to enhance the 3. And also there will certainly be 4 that match the 7. The number of factors = 3 \$ imes\$ 2 \$ imes\$ 4 = 24.

217503 = 111\$ imes\$133\$ imes\$32So there will certainly be 2 rows in the an initial column which complement the 11. Likewise in the 2nd column there will be 4 rows to enhance the 13. And also there will be 3 that match the 3. The variety of factors = 2 \$ imes\$ 4 \$ imes\$ 3 = 24.

312500 = 57 \$ imes\$ 22So there will certainly be 8 rows in the first column which enhance the 7. Likewise in the 2nd column there will certainly be 3 rows to enhance the 2. The variety of factors = 8 \$ imes\$ 3 = 24.

690625 = 171\$ imes\$131\$ imes\$ 55So there will certainly be 2 rows in the an initial column which match the 17. Likewise in the second column there will be 2 rows to complement the 13. And there will be 6 that complement the 5. The number of factors = 6 \$ imes\$ 2 \$ imes\$ 2 = 24.

94143178827 = 323So there will certainly be 24 rows in the first column which match the 3. The variety of factors = 24 \$ imes\$ 1 = 24.

Sanika and also Amy found the the smallest numbers through 14 and 15 factors. This is Amy"s work:

To discover the smallest number with 14 factors, you would certainly list all of the components of 14:2 \$ imes\$ 71 \$ imes\$ 14Then, you would pick the pair of number that have the least difference between them. In this case, the is 2 and also 7. Then take 1 away from every of the numbers:2\$-\$1 = 17\$-\$1 = 6These numbers will be the powers. Then, you would pick the 2 smallest element numbers, however, they should be different else the number of factors will be incorrect. The two prime numbers that you would should use space 2 and also 3. Then, friend would offer the smallest prime number (2), the biggest power (6).You must do this in order to ensure the you gain the smallest number possible.So, the number would certainly be:26 \$ imes\$ 31 = 192

This method would be the same to job-related out the smallest number of factors for any type of number, all that you need to change would be to swap the number 14 with the number of factors the you want your number to have. For example,to find the smallest number through 15 factors:Factors of 15:3 \$ imes\$ 5 1 \$ imes\$ 15Choose the pair of determinants with the least difference:3 and 5Take far 1:3\$-\$1 = 25\$-\$1 = 4Smallest, different, prime numbers:2 and also 3Powers:24 \$ imes\$ 3224 \$ imes\$ 32 = 144

This is Sanika"s work-related for a number v 18 factors:

To discover numbers with specifically 18 determinants we need to make certain the product of every the index number (+1) the the prime determinants give united state 18. The components of 18 include (1,18) (2,9) (3,6). We can more prime factorise the last two pairs to offer us (2, 3, 3). Smaller the exponent the lesser the worth is going come be. For this reason we will certainly take (2, 3, 3) as the exponents. We will certainly take thefirst 3 primes together bases i.e. 2 3 & 5. This will give us a last answer that 22\$ imes\$32\$ imes\$5. This pipeline us v 180.

Sanika likewise explained i beg your pardon numbers have an odd number of factors:

Numbers which are perfect squares will have an odd variety of factors together one number will be recurring whilst writing down its factor pairs.E.g. Let"s take into consideration 16= 24 \$ ightarrow\$ (1, 16) (2, 8) (4, 4)So 16 has a complete of 5 determinants as 4 is repetitive twice. This holds true because that all various other perfect squares.

Sanika uncovered the smallest number with specifically 100 factors:

Prime factorising 100 gives us 2\$ imes\$2\$ imes\$5\$ imes\$5, currently we can assign the smallest prime bases come the biggest powers, which will give us 24\$ imes34\$ imes\$5\$ imes\$7 = 45360.

See more: Colombe Jacobsen-Derstine Mib 2 002), Colombe Jacobsen

Sanika and also Amy both said that the number under 1000 with the most components is 840 (32 factors). This is Amy"s work:

Finally, to discover the number less than 1000 that has actually the many factors, you have to try and multiply together the widest range of element numbers, beginning with the lowest:2 \$ imes\$ 3 \$ imes\$ 5 \$ imes\$ 7 \$ imes\$ 11 \$gt\$ 1000 but,2 \$ imes\$ 3 \$ imes\$ 5 \$ imes\$ 7 \$lt\$ 1000 so these are the numbers the we will use.2 \$ imes\$ 3 \$ imes\$ 5 \$ imes\$ 7 = 210, so, currently we can add powers come the number 2 until the number is as close come 1000 as possible.Therefore, the answer is 23 \$ imes\$ 31 \$ imes\$ 51 \$ imes\$ 71 = 840: this is the closest number to 1000.Using the info above, the number of factors that it has actually must be 32.