Old Mac Donald Had an Impossible Farm Quiz

Answer: OX

You could try and guess this one: there aren't that many animals with a name of only two letters, and the one you'll probably find on a farm is an ox.
But let's solve this mathematically, so you could get the feeling of this technique.
First of all, you'll have to break the product in prime factors. In this case, you'll get 360 = 2*2*2*3*3*5.
Available letters are thus A(1), B(2), C(3), D(2*2=4), E(5), F(2*3=6), H (2*2*2=8), I(3*3=9), J(2*5=10), L(2*2*3=12), O(3*5=15), R(2*3*3=18), T(2*2*5=20) or X(2*2*2*3=24).
Vowels are A (1), E (5), I (9), O (15) and U (21), and sometimes you could use a Y (25) as a vowel too.
360 can't be divided into whole numbers by 21 nor 25. Dividing 360 by 1 , by 5 or by 9 would result in the quotient (respectively 360, 72 or 40) being more than the number of letters in the alphabet, so the vowel is not A, not E nor an I. That leaves us with the only possible vowel: O.
360 divided by 15 gives 24, so the other letter has to be X.
Now use the letters O and X to make a word and you'll find OX.

Answer: CAT

Breaking down 60 into prime factors, gives us 2*2*3*5. The letters which can be used are A (1), B (2), C (3), D (2*2), E (5), F (2*3), J (2*5), L (2*2*3), O (3*5) or T (2*2*5). So you'd have to pick three letters from the list A B C D E F J L O T. The solution is not quite evident, you'll need to continue to calculate.
Find the vowel to use: it's either A or E or O.
Let's try a word with O. The remaining letters could be BB, or DA. So we could try for instance BOB - but that's not an animal. ODA (or any anagram of these letters) won't do either.
Could we use the E? Remaining letters could then be AL or DC. Neither combination (ALE or anagrams, nor DEC or anagrams) will be regarded as a farm animal.
So we have only an A as vowel. Remaining letters could be CT, DO (but this one we have already discarded), EL (which also was discarded) or FJ.
AFJ or its anagrams can't form a farm animal.
So it has to be the combination ACT - anagrammed into CAT.

Answer: ASS

The clue won't give it quite away, so we'll have to try and solve it in a mathematical way.
Start factoring 361. It can't be divided into whole numbers by 2, 3, 5. Let's look for other prime numbers: neither 7 nor 11 nor 13 are integer divisors of 361. Do you think it's getting hard? Don't give up, the solution is near.
Dividing 361 by 17 won't render an integer number. Let's try 19: oh, well, 361 is the square of 19.
So the product is 19*19.
19 is the number corresponding to the letter S, so we need two S's.
But there are three letters involved. The only one we could combine is A. 361 = 1*19*19, and thus the result is ASS. And indeed, an ass is a farm animal (at least in some farms).
I'll spare you the full explanation why a donkey would give 1,155,000. If you feel up to it, try and calculate this one out - but you might need several hours to crack the code.

Answer: DOG

Once again, start breaking up the product into prime factors. 420 = 2*2*3*5*7.
Possible letters would be A (1), B (2), C (3), D (4), E (5), F (6), G (7), J (10), L (12), N (14), O (15), T (20) or U (21). Picking three letters from this list (A B C D E F G J L N O T U) is not that easy, you'll have to calculate some possibilities.
Let's first try out the combinations with U. The remaining letters could be AT, BJ or DE. So we would have either UAT, or UBJ, or UDE (each time with its anagrams). Not one of these combinations would result in a farm animal.
If we try our hand at the combinations with O, the remaining letters could be either BN or DG. Now OBN can't be anagrammed into a farm animal, but ODG is the anagram of DOG, which fits perfectly.
You could verify your answer with the combinations for E. Remaining letters are DU (already discarded), FN or LG. Solutions EFN or ELG won't give any farm animals.
Another verification is for the combinations with A. The remaining letters should be TU, and that has already been discarded. 

Answer: HEN

Well, let's try and solve this one mathematically.
Start factoring 560 into prime numbers: 560 =2*2*2*2*5*7.
So the available letters are A (1), B (2), D (4), E (5), G (7), H (8), J (10), N (14), P (16) or T (20).
The available vowels are only A and E.
If we use an A, there are only 2 letters left to multiply into 560. The closest result would be 20*28, but there are not that many letters in the English alphabet.
So the word will evidently contain an E. The remaining letters could be HN or PG.
All right, EPG won't anagram into a farm animal. EHN however can be scrambled into a HEN, and a hen is a farm animal that does lay eggs. So the correct answer is HEN.
Incidentally, the very brave of you could try and solve 498,960 and find this is a CHICKEN. However, this would take us too long to explain.

Answer: PIG

Now this one is beginning to become quite difficult if you can't guess from the clue.
Let's factor 1008 into prime numbers. 1008 = 2*2*2*2*3*3*7. (Take your time to calculate this out, you'll need several steps).
Fine, now we'll try and find the letters which can be included in this product. We could use A, B, C, D, F, G, H, I, L, N, P, R, U or X.
The available vowels are A, I and U. However, the A doesn't fit in: with two letters remaining, we can't get a product of over 1000.
Let's try the U: remaining letters would be BX, DL or FH. But neither UBX nor UDL or UFH are anagrams of farm animals.
So we need the I. Remaining letters could be PG or HN. IPG is a scramble of PIG, while rearranging the letters IHN would not give you any farm animal.

Answer: LAMB

Let's calculate this out:
Factoring 312 into prime numbers gives us 312=2*2*2*3*13.
This results in the available letters A, B, C, D, F, H, L, M, X or Z.
So the vowel we need to use is the A.
The number 13 can be only used for the M or the Z. So let's combine the A with either M or Z, for one of these letters has to be present.
AZ will lead to the remaining letters AL, BF or CD. The solution would then be a scramble of either AZAL, AZBF or AZCD. This won't give the appropriate result: none of these combinations can be scrambled into a farm animal.
So what with the combination AM? Remaining letters could then be AX, BL, CH or DF. Possible solutions would then be scrambles of AMAX, AMBL, AMCH or AMDF.
Out of these four combinations, only AMBL can be scrambled into a young farm animal, notably a LAMB.

Answer: GOAT

Can you calculate this one out?
Once again, start factoring. Eventually you'll find 2100 = 2*2*3*5*5*7.
Available letters would thus be A, B, C, D, E, F, G, J, L, N, O, T, U, Y.
Now we'll try and find the correct vowels to use, out of A - E - O - U and perhaps even Y.
Let's pick any two of these vowels and see what we can get.
AE would have to be combined with TU. Now this would give three vowels, and not a farm animal to be scrambled out of AETU.
AU leaves us with either ET (already discarded) or DY. AUDY will not scramble into a farm animal.
AY could be combined with DU (already discarded) or with GL. Now GLAY is not a scramble for a farm animal.
EO can be combined with DG or with BN. Neither EODG (scramble of the Doge) nor EOBN (scramble of Bone) can be rearranged into any farm animal.
EU leaves us with AT (already discarded), BJ or DE. EUBJ nor EUDE can be scrambled into a farm animal.
EY and OY are impossible: that would take three factors of 5.
OU is impossible too: that would take two factors of 3.
UY would leave us with AD (already discarded) or BB. UYBB is not a scramble of a farm animal.
So it has to be AO. Remaining letters would be GT or NJ. AONJ doesn't fit, but unscrambling AOGT gives us a GOAT.

Answer: DUCK

This one is rather tricky. Start factoring and you'll find 2772 = 2*2*3*3*7*11.
Available letters are A, B, C, D, F, I, K, L, N, R, U, V.
The vowel can be A, I or U.
The number eleven can give us only two possible letters too: either K or V.
So we could try our hand with AK, AV, IK, IV, UK or UV.
If we use AK, the remaining letters could be LU (this gives us two vowels) or NR. Neither AKLU nor AKNR can be unscrambled into a farm animal.
AV leaves us with either GR or FU. Neither AVGR nor AVFU can be unscrambled as we'd like.
IK gives us the remaining letters DG or BN. Neither IKDG nor IKBN unscramble as desired.
IV leaves us with BG or AN. IVBG nor IVAN satisfy our need for something that can be unscrambled into a farm animal.
UV would leave us with BC or AF. UVAF was already discarded, and UVBC will not unscramble into any English word.
So the only possible combination is UK either with DC or with BF. UKBF doesn't give what we're looking for, but UKDC unscrambles into a DUCK. Quack, quack, you've found it!

Answer: LLAMA

Can you calculate this one out? Start factoring: 1872 = 2*2*2*2*3*3*13.
Possible letters are thus A, B, D, F, H, I, L, M, P, R, Z.
So the vowels would be either A or I.
We need the same vowel twice, so we'll have to discard the I: there aren't enough factors of 3 to have the letter I twice.
Thus the vowels are AA, and no I is present. What's left?
We can use the factor 13 for two letters only: either M or Z.
AAZ leaves us with CX, DR or FL (as we can't use the letter I). Neither AAZCX, nor AAZDR nor AAZFL unscramble into a farm animal - however unusual it might be.
AAM has to be combined with HR or LL, as we can't use the letter I. AAMHR doesn't quite fit our needs, but AAMLL will unscramble into LLAMA.