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Let a < b < c < d be four consecutive positive integers such that

- a is divisible by 5,
- b is divisible by 7,
- c is divisible by 9, and
- d is divisible by 11.

Find the minimum value of a+b+c+d.

Jul 3, 2020

#1
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Let a < b < c < d be four consecutive positive integers such that

- a is divisible by 5,
- b is divisible by 7,
- c is divisible by 9, and
- d is divisible by 11.

Find the minimum value of a+b+c+d.

$$\begin{array}{ll} \left.\begin{array}{rcl} a & \equiv & 0 \pmod{5} \\ b=a+1 & \equiv & 0 \pmod{7} \\ c=a+2 & \equiv & 0 \pmod{9} \\ d=a+3 & \equiv & 0 \pmod{11} \end{array}\right\} \begin{array}{rcl} a & \equiv & 0 \pmod{5} \\ a & \equiv & -1 \pmod{7} \\ a & \equiv & -2 \pmod{9} \\ a & \equiv & -3 \pmod{11} \end{array} \\\\ a+b+c+d = 4a+6 \\ \end{array}$$

$$\begin{array}{ll} \left.\begin{array}{rcl} a & \equiv & 0 \pmod{5} \\ a & \equiv & -1 \pmod{7} \\ \end{array}\right\} \begin{array}{rcl} a & \equiv & -15 \pmod{35} \\ \end{array} \\ \left.\begin{array}{rcl} a & \equiv & -2 \pmod{9} \\ a & \equiv & -3 \pmod{11} \end{array}\right\} \begin{array}{rcl} a & \equiv & -47 \pmod{99} \\ \end{array} \end{array}$$

$$\begin{array}{ll} \left.\begin{array}{rcl} a & \equiv & -15 \pmod{35} \\ a & \equiv & -47 \pmod{99} \\ \end{array}\right\} \begin{array}{rcl} a & \equiv & 1735 \pmod{3465} \\ a &=& 1735 + 3465n,\ n\in \mathbb{Z} \\ \end{array} \\ \end{array}$$

$$\text{The minimum value of \mathbf{a} is \mathbf{1735}} \\ \text{The minimum value of \mathbf{a+b+c+d} = 6*1735 + 4 \mathbf{=6946} }$$

Check:

$$\begin{array}{|rcll|} \hline 1735 & \equiv & 0 \pmod{5} \\ 1736 & \equiv & 0 \pmod{7} \\ 1737 & \equiv & 0 \pmod{9} \\ 1738 & \equiv & 0 \pmod{11} \\ \hline \end{array}$$

Jul 3, 2020