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A, B and C, in that order, are three-consecutive whole numbers. Each is greater that 2000. A is a multiple of 4. B is a multiple 5. C is a multiple of 6. What is the smallest possible value of A?

Sagot :

Corsaquix

Answer:

[tex]A=2044[/tex]

Step-by-step explanation:

Note that [tex]x\in\mathbb{W}[/tex] denotes that [tex]x[/tex] is a whole number.

By definition, consecutive numbers follow each other when we count up (e.g. 1, 2, 3).

Let's consider our conditions:

  • A, B, and C are consecutive whole numbers greater than 2,000
  • A is a multiple of 4
  • B is a multiple of 5
  • C is a multiple of 6

Since B is a multiple of 5, the ones digit of B must be either 0 or 5. However, notice that the number before it, A, needs to be a multiple of 4. The ones digit of a number preceding a ones digit of 0 is 9. There are no multiples of 4 that have a ones digit of 9 and therefore the ones digit of B must be 5.

Because of this, we've identified that the ones digit of A, B, and C must be 4, 5, and 6 respectively.

We can continue making progress by trying to identify the smallest possible whole number greater than 2,000 with a units digit of 6 that is divisible by 6. Notice that:

[tex]2000=2\mod6[/tex]

Therefore, [tex]2000-2=1998[/tex] must be divisible by 6. To achieve a units digit of 6, we need to add a number with a units digit of 8 to 1,998 (since 8+8 has a units digit of 6).

The smallest multiple of 6 that has a units digit of 8 is 18. Check to see if this works:

[tex]C=1998+18=2016[/tex]

Following the conditions given in the problem, the following must be true:

[tex]A\in \mathbb{W},\\B\in \mathbb{W},\\C\in \mathbb{W},\\A+1=B=C-1,\\A=0\mod 4,\\B=0\mod 5,\\C=0\mod 6,[/tex]

For [tex]C=2016[/tex], we have [tex]B=2015[/tex] and [tex]A=2014[/tex]:

[tex]A\in \mathbb{W},\checkmark\\B\in \mathbb{W},\checkmark\\C\in \mathbb{W},\checkmark\\A+1=B=C-1,\checkmark\\A=2014\neq 0\mod 6, \times\\B=2015=0\mod 5,\checkmark\\C=2016=0\mod 6\checkmark\\[/tex]

Not all conditions are met, hence this does not work. The next multiple of 6 that has a units digit of 8 is 48. Adding 48 to 1,998, we get [tex]C=1998+48=2046[/tex].

For [tex]C=2046[/tex], we have [tex]B=2045[/tex] and [tex]A=2044[/tex]. Checking to see if this works:

[tex]A\in \mathbb{W},\checkmark\\B\in \mathbb{W},\checkmark\\C\in \mathbb{W},\checkmark\\A+1=B=C-1,\checkmark\\A=2044=0\mod 4,\checkmark\\B=2045=0\mod 5,\checkmark\\C=2046=0\mod 6\checkmark[/tex]

All conditions are met and therefore our answer is [tex]\boxed{2,044}[/tex]

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