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You asked for intuitive sense and I'll try to provide it. The formula is:

$$\Delta S = \frac{\Delta Q}{T}$$

So, you can have $\Delta S_1=\frac{\Delta Q}{T_{lower}}$ and $\Delta S_2=\frac{\Delta Q}{T_{higher}}$

Assume the $\Delta Q$ is the same in each case. The denominator controls the "largeness" of the $\Delta S$.

Therefore, $\Delta S_1 > \Delta S_2$