Math Question!!!?

Question

Remnok the Nomad has a brother, Damon the Nomad, who's fond of coming up with math puzzles. Anyway, he came up with one that uses his own name:

Assume each letter in the following formula represents a unique digit: (for example, ABCD could be the number 1234)

NOMAD x 4 = DAMON

What is the number represented by the digits DAMON?

Answer 1

NOMAD
x4
======
DAMON

Looking at the 5th digit column we find that since N is the units digit of something multiplied by 4,it must be even. But since DAMON and NOMAD have the same number of digits, we know that N * 4 < 10 ... or N<3. So N must be 2.

Looking If N=2, going back to the first digit we find that D must be 8 or 9 (depending on if the second digit column carries). But since we know that D *4 ends in 2, D must be 8 (9*4 ends in 6).

So we have:

2OMA8
x4
======
8AMO2

Now since D=8, we know that the 2nd digit column did not carry, so O * 4 < 10. Therefore O is 0,1,or 2. Butlooking at the fourth column we find that O must be odd, as it had three carried over from the fifth column (8*4 = 32) and that three is added to an even number (the last digit of A * 4 would be even). This means O must be the only odd number in {0,1,2} . So O = 1

Looking at the fourth digit column, this means that A*4 ends in 8 (as when the 3 is carried over the result is 1). The only two choices that when multiplied by 4 result in 8 are 2 or 7. But we've already used 2 for N. Therefore A = 7.

21M78
x4
======
87M12

We just need to figure out the 3rd digit column. The column has 3 carried into it (78 *4 = 312), It also carries out 3 (87 = 21 *4 + 3).

So:

M * 4 + 3 = 30 + M
3M = 27
M=9

21978
x4
======
87912

The number represented by DAMON is 87912.

Answer 2

Solutions:

Since NOMAD is a five-digit number, N is not equal to 0.
Since the product has at most 5 digits then in the multiplicand N = 1 or 2;
..........and in the product D is greater than or equal to 4.

From the units place, 4 x D is even, therefore with "N = 1 or 2" (as previously mentioned), then N = 2.
..........which in turn implies that D = 3 or 8.

This, together with "D is greater than or equal to 4" (as previously mentioned), implies that D = 8.

So,

.......3
2 O M A 8
..........x 4
________
8 A M O 2

Looking at the tens place, O is odd. And looking at the thousands place, O is 1 or 2 (no "carrying" in the thousands place). But because O is odd, then O = 1.

...b a 3
2 1 M A 8
..........x 4
________
8 A M 1 2

From the tens places A = 2 or 7.
On the other hand, in the hundreds place b = 0, 0, 2 or 3; so that A is not equal to 2. Therefore A = 7.

...3 3 3
2 1 M 7 8
.........x 4
________
8 7 M 1 2

In the hundreds place, since 3 is "carried" to the thousands place, M must be a digit such as (4 x M) + 3 is greater than 30. So M must be 7, 8 or 9. We use trial and error to find the value of M, and found that only M = 9 satisfies the multiplication. So NOMAD stands for 21978.

...3 3 3
2 1 9 7 8
........x 4
________
8 7 9 1 2

I remember I already answered this kind of question before... o.O hmmm.

Answer 3

4*(0,1,2,3,4,5,6,7,8,9)=(0,4,8,2,6,0,4,8,2,6) (mod 10)
so N is in {0,4,8,2,6}
We can eliminate 0 because Nomad wouldn't really be a 5 digit number
We can eliminate {4,8,6} becuase *4 would give another digit
Therefore N=2
4*D ends with N so D is 3 or 8
but 4*2OMA8=DAMO2 so D can't be 3
so D is 8 and 4*OMA8 < 10000 (no carry)
OMA8 < 2500 and O is not 2, so O must be 0 or1
4*2OMA8=8AMO2 has a carry from 4*8 so O can not be 0, so it must be 1
4*21MA8=8AM12
in the second place we carried 3 from 4*8, so before the carry we must have had 8 so A must be 2 or 7. A is not 2 so it must be 7
4*21M78=87M12
now 4*21078 = 84312, so we need 4M to be 32 or 36
We have already used 8, so M must be 9
4*21978 = 87912

Answer 4

DAMON DIVIDED BY FOUR
DAMON/4

Answer 5

21978 x 4 = 87912

D = 8
A = 7
M = 9
O = 1
N = 2

DAMON = 87912