I definitely wouldn't recommend it.Hamsters have specific nutritional needs and very few US hamster foods actually meet those needs.It's an investment to get appropriate food,but in the end feeding only the highest quality food and treats(and certain supplements)can save you money on vet bills later on.
One option that may need to be done due to a lack of Harry/Hazel Hamster being sold in Petco(not sure when this changed)would be Higgins Sunburst hamster/gerbil food and Carefresh hamster food.These two on their own wouldn't be very good,but in a two thirds Carefresh to one third Higgins ratio they're just a tiny bit under the minimum nutritional requirements of hamsters.However,a simple addition of freeze dried chicken and a pinch of plain flax seed is all that is needed to make it suitable for Syrians
Make sure no treats or foods ever contain artificial flavors,artificial colors,unspecified biproducts(digest,derivatives,meat and bone meal of any sort,biproduct meals,animal fat or otherwise unspecified fat products,animal liver or otherwise unspecified liver products,etc),unspecified preservatives or artificial preservatives(especially BHA/BHT/ethoxyquin which are known to cause cancer,keep in mind if you ever buy UK products that these are often hidden behind the "E&C approved antioxidants/preservatives"loophole and therefore UK products with this terminology must not be used),artificial sweeteners or processed sugars(dextrose,sorbitol,sucrose,fructose,maltodex trin,etc),propylene glycol,or sulfur dioxide.Nothing should ever be made in China or irradiated either,and if it is a product you may share with dogs or cats it must not have corn,wheat,or wheat products/wheat relatives(safe for hamsters,terrible for dogs and cats,some examples of those products/relatives would be wheat germ and spelt).