How many 4-digit positive integers are multiple of each integer from

Question

How many 4-digit positive integers are multiple of each integer from 1 to 10?

A. 1
B. 2
C. 3
D. 4
E. 5

Bunuel · Accepted Answer

4-digit positive integer should be multiple of: 2, 3, 4(=2^2), 5, 6(=2*3), 7, 8(=2^3), 9(=3^2), 10(=2*5). Basically our 4-digit integer should be multiple of LCM of these numbers, which is  2^3*3^2*5*7=2520 .

There are 3 such numbers: 2520, 5040, and 7560.

gnus · Answer

I am going to assume that you mean 1 to 10 inclusive.

If you wanted to know if a number was divisible by all those digits, you would have to find the smallest # that was divisible by 1 through 10.

1- Every # from 1000 to 9999
1 to 2 - Every even number
1 to 3 - Every multiple of 6 from 1000 to 9999
1 to 4 - Every multiple of 12 from 1000 to 9999
1 to 5 - Every multiple of 60 From 1000 to 9999
1 to 6 - Every multiple of 60 from 1000 to 9999
1 to 7 - Every multiple of 210 from 1000 to 9999
1 to 8 - Every multiple of 840 from 1000 to 9999
1 to 9 - Every multiple of 2520 from 1000 to 9999
1 to 10 - Every multiple of 2520 from 1000 to 9999

With that said, there are three 4-digit numbers that satisfy the requirements of being divisible by 1 through 10 inclusive.

gurpreetsingh · Answer

didnt get u.....pls elaborate

gnus · Answer

If you mean you didn't get my response. Here is the breakdown:

To get a number that is divisible by all 10 numbers you could do 10!. But 10! = 3628800 and it is not the lower common multiple. To get the LCM, I took all the numbers like this:
1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10.
I removed the 6, 8, 9, 10 and added a 3.
I ended up with 1 X 2 X 3 X 3 X 4 X 5 X 7 which equals 2520
I removed the 6 because it is a multiple of 2 and 3 earlier in the list.
I removed the 8 because it is a multiple of 2 and 4 earlier in the list.
I removed the 10 because it is a multiple of 2 and 5 earlier in the list.
For the 9 to be removed, you needed two 3's. So I added a 3 and removed the 9.

With those adjustments I came up with 2520 as the lowest common multiple for integers from 1 to 10.

Because the question asked between the range of 1000 and 9999, three numbers satisfy the requirements: 2520, 5040, 7560.

Finding the lowest common multiple can be a little confusing. I didn't know how to calculate it so I had to work out how to do this on a smaller subset. I used the one through five range to figure how to figure out the LCM.

TheSituation · Answer

Fabulous approach and solution. Just goes to show that no matter how much you go :shock:

when you first read a question that there is almost always a shorter route to the answer, if you are clever enough.

BarneyStinson · Answer

You nailed the secret to approach GMAT!!! In fact, you are already a Manager. Focus, problem, possibilities and solution, that is all it is in that big bad corporate world. Double Hi Fives bro!!!

TheSituation · Answer

Let's meet up at MacLaren's after work for a pint, we can discuss how we gon run dis town after we takeover.

Bunuel · Answer

The least 4-digit number which is multiple of each integer from 1 to 10 is LCM of these numbers and equals to  2^3*3^2*5*7=2520 .

Now, if we multiply this number by 2 and 3 we will still have 4-digit number which is multiple of each integer from 1 to 10 -->  2520*2=5040  and  2520*3=7560  (if we multiply by 4 the number will be 5-digit). So there are only 3 such numbers.

Hope it's clear.

mainhoon · Answer

Bunuel, awesome answer... Can you please check my post on combinations and answer it when you have a chance.. Thanks

Posted from my mobile device

PareshGmat · Answer

Find LCM of nos between 1 & 10
which also means just LCM of 7,8,9 & 10 (As all smaller are multiple of these)

Just 2 is repeated once, else all are unique

7*8*9*5 = 63*40 = 2520

2520, 2520+2520, 2520+2520+2520 is the answer

dineshnaidu2410 · Answer

How many 4-digit positive integers are multiple of each integer from 1 to 10

Least No which could be divisible by integers   is the LCM of the said integers.

LCM   = 2520

No the Nos which are divisible by 2520 will be divisible by

There are 3 values possible between 1000 to 9999

They are 2520 x 1= 2520
 2520 x 2 = 5040
 2520 x 3 = 7560

gracie · Answer

7*8*9*10=5040
5040/2=2520
2520*3=7560
3

HarshavardhanR · Answer

We need 4-digit positive integers that are multiple of every integer from 1 to 10.

- Think about how to cover every integer from 1 to 10.
- Every number, by default, is a multiple of 1.
- If a number is divisible by 8, 9, 5, and 7, then, it is also divisible by 2,3,4,6, and 10.
- Hence, we only need to focus on finding every single 4-digit number that is divisible by 8, 9, 5, and 7 (i.e., 2^3, 3^2, 5, and 7)

The first such number = 2520 (LCM of 8, 9, 5, and 7)

There are only two other 4-digit numbers possible: 2520, 2520+2520 = 5040, 2520+2520+2520 = 7560).

Hence, the answer is 3. Choice C.

---
Harsha

egmat · Answer

When a number must be divisible by  multiple  integers, it must be a multiple of their  LCM (Least Common Multiple) .

Step 1: Find the LCM of 1 through 10 
Break each number into prime factors:
- 8 = 23(highest power of 2)
- 9 = 32 (highest power of 3)
- 5 = 5 (highest power of 5)
- 7 = 7 (highest power of 7)

LCM = 23 × 32 × 5 × 7 = 8 × 9 × 5 × 7 =  2520

Step 2: Find 4-digit multiples of 2520 
4-digit numbers range from  1000  to  9999 .

Let's list the multiples of 2520:
- 2520 × 1 =  2520   ✓ (4 digits) 
- 2520 × 2 =  5040   ✓ (4 digits) 
- 2520 × 3 =  7560   ✓ (4 digits) 
- 2520 × 4 =  10080   ✗ (5 digits - too big!)

Therefore,  there are exactly  3  four-digit positive integers that are multiples of each integer from 1 to 10.

Common Trap:   Trying to check divisibility by each number (1, 2, 3... 10) separately is time-consuming and error-prone. Always use LCM for "divisible by all" questions!

Answer: C

How many 4-digit positive integers are multiple of each integer from

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