If there is inequality in the pattern of population growth, there is greater inequality in food production and utilisation. As societies become wealthier, their consumption of animal products increases. This means that a greater proportion of such basic foodstuff as grains and legumes that could feed humans directly is instead being converted into feed for poultry and large farm animals. Yet this conversion of plant-based food into animal food for humans is far from efficient. Only $16 \%$ of the calories fed to chickens are recovered by us when we eat them. This conversion rate goes down to five to seven per cent in large animals that are fed grain to add fat and some protein before slaughter.

What is the maximum value of $n$ such that $7 \times 343 \times 385 \times 1000 \times 2401 \times 77777$ is divisible by $35^{\mathrm{n}}$ ?

If there is inequality in the pattern of population growth, there is greater inequality in food production and utilisation. As societies become wealthier, their consumption of animal products increases. This means that a greater proportion of such basic foodstuff as grains and legumes that could feed humans directly is instead being converted into feed for poultry and large farm animals. Yet this conversion of plant-based food into animal food for humans is far from efficient. Only $16 \%$ of the calories fed to chickens are recovered by us when we eat them. This conversion rate goes down to five to seven per cent in large animals that are fed grain to add fat and some protein before slaughter.

If $N^2=12345678987654321$, then how many digits does the number N have?

In only 50 years, the world's consumption of raw materials has nearly quadrupled, to more than 100 billion tons. Less than $9 \%$ of this is reused. Batteries of old vehicles contain materials such as lithium, cobalt, manganese and nickel that are pricey and can be hard to obtain. Supply chains are long and complicated. Buyers' risks are being aggravated by their suppliers' poor environmental and labour standards. Reusing materials makes sense. Once batteries reach the ends of their lives, they should go back to a factory where their ingredients can be recovered and put into new batteries.

If $n$ is a natural number, then what is the number of distinct remainders of $\left(1^n+2^n\right)$ when divided by 4 ?

In only 50 years, the world's consumption of raw materials has nearly quadrupled, to more than 100 billion tons. Less than $9 \%$ of this is reused. Batteries of old vehicles contain materials such as lithium, cobalt, manganese and nickel that are pricey and can be hard to obtain. Supply chains are long and complicated. Buyers' risks are being aggravated by their suppliers' poor environmental and labour standards. Reusing materials makes sense. Once batteries reach the ends of their lives, they should go back to a factory where their ingredients can be recovered and put into new batteries.

What is the 489 th digit in the number $123456789101112 \ldots$ ?